Thursday, January 22, 2015

Links of interest


Has mathematics misled modern science?  Bryan Appleyard, channeling physicist Lee Smolin and philosopher Roberto Mangabeira Unger, makes the case.

But maybe mathematical elegance should trump empirical evidence?  Some physicists seem to think so.  In Nature, physicists George Ellis and Joe Silk will have none of it.  Further commentary, and a roundup of other responses, from physicist Peter Woit

At the OUP Blog, John Searle on the intentionality of perceptual experience.  At the same blog: Federica Russo and Phyllis Illari on causation in science and Tad Schmaltz on causation in Aristotle and Hume.

Philosopher John Lamont on Thomism, “manualism,” and the nouvelle th√©ologie, at Rorate Caeli

At The Catholic Thing: Frank Beckwith on misguided arguments for God’s existence; Fr. James Schall on natural resources; and Gerald Russello on the debate over Catholic social teaching.

What should we think about the recent terrorist attacks in France?  At Mercantornet, Fr. Schall draws some lessons from Paris.


Books about Leo Strauss just keep appearing. Robert Howse’s new one is reviewed at The National Interest, Michael and Catherine Zuckert’s at Public Discourse, and Arthur Melzer’s at The Week.


“Like rabbits”?  Commentary from Matthew Schmitz and Joseph Shaw.

Terence Parsons’ new book Articulating Medieval Logic is reviewed at Notre Dame Philosophical Reviews; Anna Marmodoro’s Aristotle on Perceiving Objects is also reviewed.

Want a 24 volume Neo-Scholastic theology and philosophy collection?  Bid now.

179 comments:

John West said...

Has mathematics misled modern science? Bryan Appleyard, channeling physicist Lee Smolin and philosopher Roberto Mangabeira Unger, makes the case.

I'm no physicist, but a good friend is an astrophysicist. His sentiments reflect those written about in the first link. From the way he talks, superstring theory is just a joke that gets more press than it deserves.

John West said...

Those who haven't already read it may be interested in James Franklin's recent paper: Two Caricatures I: Pascal's Wager

John West said...

Well, on a second look, maybe not that recent^

Allen Hazen said...

As to string theory being a joke…
A friend told me this one:
A string theorist's husband comes home to find her in bed with another man. She says "Dear, I can explain everything."

My friend claims he told this to a physicist who thought for a moment and said "Well, she can EXPLAIN everything… but she can't predict what's going to happen next."

Preston Cobb said...

Professor Feser,

Are you familiar with/would recommend any of the books in the Neo-Scholastic collection individually? I don't recall seeing any of them in your "Library" posts from a while back (which I've only read recently as I have only been following your blog for about a year now, and am working my way through the backlog).

P.S. Thank you for writing Aquinas and The Last Superstition, I profited from (not to mention enjoyed) them immensely and have bought copies for friends. I also have my copies of Locke and your Aristotle collection in my to read pile. And thank you for introducing me to these thinkers who many I know would reject as curiosities, but who I find offer a clearer picture of reality than I've ever seen.

seanrobsville said...

Elevating mathematics above experimental ('experienced') science is as counterproductive and messy as attempting to enclose a bottle in its contents.

Eugene Gendlin "points out that the universe (and everything in it) is implicitly more intricate than concepts, because a) it includes them, and b) all concepts and logical units are generated in a wider, more than conceptual process (which Gendlin calls implicit intricacy). This wider process is more than logical, in a way that has a number of characteristic regularities. Gendlin has shown that it is possible to refer directly to this process in the context of a given problem or situation and systematically generate new concepts and more precise logical units.

Because human beings are in an ongoing interaction with the world (they breathe, eat, and interact with others in every context and in any field in which they work), their bodies are a "knowing" which is more than conceptual and which implies further steps. Thus, it is possible for one to drive a car while carrying on an animated conversation; and it is possible for Einstein to say that he had a "feel" for his theory years before he could formulate it."
http://en.wikipedia.org/wiki/Eugene_Gendlin

John West said...

Above it for what though?

As for physics, I don't see why it's so surprising to people that physics at its furthest, bleeding theoretical edge, is less stable than at its core. That's just obvious. It should only bother someone if they're treating science like a religion, instead of like science. They're theoretical physicists, they theorize; sometimes they're wrong. When in the history of science has it ever been otherwise?

Anonymous said...

Really enjoyed John Lamont's article on neomodernism. It has helped me to understand the denigration of the " manualists " originating from the pen of an unnamed, but influential ecclesiastical source. It also makes me wonder who, among modern authors, represent the neomodernist position.

I hear echos of this position from time to time, so it seems evident that it is still influential.

Linus2nd

Anonymous said...

Thanks for posting the links on "Rabbits". As a farther of five, going on six, I found the Pope's words very discouraging. Any committed catholic who rejects the use of contraception with even a marginal fertility level will typically have a large family by modern standards using, even using NFP.

I'd be interested in hearing your thoughts about this too Ed.

Cheers,
Daniel

Scott said...

@Preston Cobb:

If you're thinking of Ed's series of "Scholastic's Bookshelf" posts, then there's some overlap but not a lot. I spotted Harper, Mercier, Boedder, and a couple by Joseph Rickaby. (Coppens made both lists, but with different books.)

Lurker said...

Hey Daniel,

If you want the best, move even handed interpretation of what the Pope says I have to strongly recommend Jimmy Akin.

Jimmy has got to be the most level headed apologist living.
Most apologists turn me off. So much so that it doesn't even seem right calling Jimmy an apologist (at least how that term gets viewed today).

Gene Callahan said...

Aargh! Searle claims Berkeley believes the exact opposite of what Berkeley actually believes!

Lurker said...

Hi Gene,
Could you spell that out a bit?

Scott said...

@Lurker:

Searle attributes to Berkeley the view that "[w]e never directly perceive objects and states of affairs in the world. All we ever perceive are the perceptual contents of our own mind." Berkeley actually held that these were at bottom the same thing: what we directly perceive really does exist, but the objects of our perception just are sensible qualities and collections thereof. They exist independently of our perception (i.e. even when we don't perceive them), and thus constitute the real natural world, because they're always perceived by God.

Ismael said...

@John West

Although I am not working in Cosmology, as a physicist, I think that what Sean Carroll and others are peddling... is appalling.

I do not wish to make any direct claims on String Theory, since I am no expert, but I think that Sean Carroll's idea to "retire the falsification principle" is idiotic.


Carroll and co. behave in some ways, like an "apocalyptic cult". They make some predictions, most unverifiable, and when, as it happens, those who can be verified (like the search for super-partners in particle physics) is not... they push the boundaries of the testability beyond of what we can do at the moment... just like a cult moving the "doomsday" to a still a future date.

---

I would also argue that thinking that math can be the only base to see if a theory is correct is ludicrous.

Even moving away from "the final frontier" of Cosmology and particle physics, if we look at science in general, we see that some phenomena can have competitive theories... but discriminate which one is correct (if any) is quite difficult even in a field where testability is much more possible than in particle physics (an example is charge transport in disordered molecular materials).


I think that if scientists who do not work in Cosmology/String Theory would make claims and behave like Sean Carroll and some of his colleagues (I do not wish to generalize!) they would become the laughing stock of their field!

---

That said I am not against the idea of using mathematics to advance physics. Math can be a wonderful tool to explore new IDEAS and (logical) possibilities… BUT at the same time common sense should not be checked at the door, hence empirical test and falsifiability ought not to be ignored as they are the cornerstone of science together with mathematics, and both are necessary for GOOD science.


I think most of these “crusades” performed by String Theorists and Cosmologists might have to do with the atheism of some of them (certainly Krauss and Carroll)… the NEED to assert their theory is correct to ”explain away God”. IRONICALLY it would NOT be explained away even IF their theories were correct (and Feser and other have discussed this at length already).

Either that or they NEED for their theory to be true… because otherwise 10, 20 or 30 years of scientific contribution would mean nothing except maybe a paragraph in a book on the history of scientific ideas, just like all the theories on the ‘ether’ or the ‘caloric fluid’. I am sure their pride cannot even accept the possibility their work is wrong.

John West said...

Ismael,

I think that if scientists who do not work in Cosmology/String Theory would make claims and behave like Sean Carroll and some of his colleagues (I do not wish to generalize!) they would become the laughing stock of their field!

As you later imply, cosmology is a data scarce field, and I think people let them get away with far more than they would others because of this.

I would also argue that thinking that math can be the only base to see if a theory is correct is ludicrous.

I think most of the philosophers of science arguing about what makes a good hypothesis would agree, though I'm not terribly well read on that subject.

That said I am not against the idea of using mathematics to advance physics. Math can be a wonderful tool to explore new IDEAS and (logical) possibilities… BUT at the same time common sense should not be checked at the door, hence empirical test and falsifiability ought not to be ignored as they are the cornerstone of science together with mathematics, and both are necessary for GOOD science.

I agree that we should approach science holistically (in the sense of considering whole systems).

Incidentally, I hold that most mathematics in some sense exists as either Platonic Forms or reflections of the Divine Nature. But I would never say that all mathematics is instantiated in the physical world, or that the physical world is entirely reducible to mathematics [1]. Cosmology is a data scarce field, but that's still no excuse for doing bad science, bad philosophy, and bad mathematics.



[1] Though, I recently had a metaphysics professor who would challenge this point. He would say physical reality is simply a reflection of the deeper, mathematical reality. I don't, however, see how his view accounts for (for example) the experience of change (the mathematical realm is completely static on most Platonic views).

seanrobsville said...

Mathematical statements and models don't explain anything because they aren't intrinsically about anything. They are isomorphic with real-world processes, but isomorphism does not entail intentionality.

Consider the statement
IF a*b > c THEN d=1

This is isomorphic with real world models of (i) A visit to a home furnishing store (ii) A succesful theatrical production.

(i) IF RoomLength * RoomWidth > CarpetArea THEN NeedMoreCarpet = TRUE

(ii) IF Audience * TicketPrice > HireOfVenue THEN AvoidedBankruptcy = TRUE

Intentionality comes from substitution of a,b,c,d,1 etc by symbolic variable names, which nevertheless have no intrinsic meaning for any computational process, and are stripped out of a computer program at compilation. They merely function as mnemonics to remind the programmer what the code is about.

Isomorphism is a one-to-one correspondence between the elements of two sets such that the result of an operation on elements of one set corresponds to the result of the analogous operation on their images in the other set. Two isomorphic processes cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. The map isn't the territory.

This distinction between isomorphism and 'aboutness' is of course at the root of the Chinese Room argument, and also probably at the root of the Hard Problem,where there is an explanatory gap between neural states, which are presumably capable of being modelled isomorphically by computer systems, and their experiential correlates.

Intentionality comes from the side of the observer, not from the system. You can't make a mathematical model of the mathematician.

Anonymous said...

Wrong thread, perhaps?

Irish Thomist said...

I'll be both as frank and as blunt as this deserves in relation to two links about what the Pope said. Can people just stop it with the Pope bashing? Seriously. You can't blame the Pope for how people knowingly quote him out of context.

A charitable reading isn't what is needed, just reading what he actually said - which I take the time to do - is enough.

It is the media using his pastoral style, not Francis saying anything wrong.

Irish Thomist said...

@Daniel

It was clear from what the Pope said that there is misconception among some that we need to make as many children as possible regardless of the situation.

This is not Church teaching, not in Humanae Vitae, not in what John Paul II or anyone else said on the matter.

I am not sure how much detail Love and Responsibility goes into the issue in.

http://www.ewtn.com/vexperts/showmessage.asp?number=302814

Talks about it. Although I will not say I know enough to agree or disagree about the theology.

Glenn said...

(Hey Sean,

(IF RoomLength * RoomWidth > CarpetArea THEN NeedMoreCarpet = TRUE

(What company do you work for? The Home Depot?

(1. I want to carpet the area in front of the fireplace.
(2. The area to be carpeted measures 2 feet by 5 feet.
(3. Yesterday I purchased a 3'x 8' carpet remnant.
(4. Today I measured the room where the fireplace is located, and found it is 50' by 20'.

(Question: Do I need more carpet?

(Answer: According to your formula -- previously presented in May of '12 (Kripke contra computationalism), and again in November of '12 (Nagel and his critics, Part IV) -- the answer is an unqualified yes.

(PS I measured the room because I'll be painting it next month, and want to figure how much paint I'll need to buy. Have you any handy-dandy, 'isomorphic with real-world processes' paint formulas you'd like to share? I ask only because wasting money is a hobby of mine.

(Btw, sorry (in a social, white-lie kind of way) for the sarcasm. However, it was pointed out 2+ years ago that the formula fails, and now you have one reason why.)

Scott said...

@seanrobsville:

"Mathematical statements and models don't explain anything because they aren't intrinsically about anything."

If this is meant as a blanket denial that mathematics has any explanatory power, then it strikes me as wildly wrong-headed. Perhaps you mean only that the symbols or sounds used in a mathematical statement have no intrinsic intentionality, or that the steps in a computer program (which aren't "mathematical statements" at all, and are "instructions" only via derived intentionality) aren't "intrinsically about anything." If either of those is what you mean, then you'll get no argument from me.

But mathematical statements are most certainly "intrinsically about" something. Every statement is "intrinsically about" something, and so, a fortiori, is every mathematical statement. What would make it a "statement" otherwise?

And surely mathematical statements explain some things. If I know, for example, that any three points in Euclidean 3-space are coplanar but four need not be, then I have at least a good start on an explanation of why three-legged stools don't wobble but four-legged ones often do.

Scott said...

(Any three noncollinear points.)

John West said...

Mathematical statements and models don't explain anything because they aren't intrinsically about anything

Also, if any full-blooded, plenitudinous realism about uninstantiated mathematics is the case, then I'm unsure how mathematical statements aren't intrinsically about anything.

As for how we would know, on a plenitudinous view, all we would have to do is prove consistency (ie. Balaguer's externalism).

Scott said...

[slaps forehead again] And of course three collinear points are coplanar as well; they just don't uniquely determine a plane, which of course is what I meant.

Ah, well.

John West said...

I should add to my previous comment, that I think Divine Exemplarism may even be necessarily plenitudinous thanks to divine omnipotence.

Irish Thomist said...

@seanrobsville

Sure in some sense the application of mathematics qua a representational system of some logic that seems to be intrinsic in the universe aren't about anything if we abstract them out entirely from that which they are referring to. As a blanket statement however I would be careful.

It's true we come to know mathematical truth through observation and reason initially, it is equally true that we must presuppose the truth of mathematics (of some kind) for any further experimental or empirical assessment of things in greater detail.

Have I misunderstood or are you really saying experimental methodology comes before the math it requires in epistemic priority? Maybe I have skimmed over what you said too quickly or something.

John West said...

Irish Thomist,

Sure in some sense the application of mathematics qua a representational system of some logic that seems to be intrinsic in the universe aren't about anything if we abstract them out entirely from that which they are referring to. As a blanket statement however I would be careful.

But given a plenitudinous realism, any possible consistent mathematical statement would be intrinsically about something; namely, the corresponding Idea.

Have I misunderstood or are you really saying experimental methodology comes before the math it requires in epistemic priority?

This would be true on the Scholastic view, but that doesn't mean that it would have to be us doing the abstracting. Rather, the abstracting from nature (required to get us started with doing math) could have been done by others before us (Greek geometers, etc.). They could have then distilled that into a body of knowledge that has been passed down and added to with each generation, as the discipline of mathematics.

This could explain how even though on the Scholastic view mathematics must be abstracted from nature, abstracting from nature isn't at all the way mathematicians work (and once we get started, all we need to know is that it's consistent to know it corresponds to some Idea).

John West said...

Figuring out which mathematics are instantiated in the physical part of the world would, of course, require empirical investigation.)

Timocrates said...

America Magazine also makes some good (albeit controversial) points in regards to the Charlie Hebdo affair and sacred cows:

http://americamagazine.org/content/dispatches/charlie-hebdo-and-sacred-cows-place-mockery-civil-society

Timocrates said...

Also this is a very well done round-up Professor Feser! It's worthy in its own right and these might deserve a special place as I can see people coming to read the contents of this round-up for its own sake! You might want to speak with some young hipster website-savvy and internet marketing kids. I would also consider adding a forum or maybe making a new webpage that includes this blog as the comment sections are getting so popular that a forum is almost necessary to keep track of the important debate being generated by your articles.

John West said...

Not to sound like an old curmudgeon, but "It's worked fine the way it is for years. Why change it?"

Timocrates said...

And here's a gem comment I would like to share with other Feser fans from Mr. Sakker's blog:

I wrote,
"We cannot prove that a being that is what it is, is not what it in fact actually is!"

A commenter on Mr. Sakker's blog replied:

"That’s because that is circular logic, Timo. It is what it is because…it is what it is."

We have a long way to go and a lot of work to do with our eliminative materialist friends!

Timocrates said...

Islam is not at war with or stirring up hatred against Christianity.

"“We shall unleash the nihilists and the atheists and we shall provoke a great social cataclysm which in all its horror will show clearly to all nations the effect of absolute atheism; the origins of savagery and of most bloody turmoil.

Then everywhere, the people will be forced to defend themselves against the world minority of the world revolutionaries and will exterminate those destroyers of civilization and the multitudes disillusioned with Christianity whose spirits will be from that moment without direction and leadership and anxious for an ideal, but without knowledge where to send its adoration..."


- U.S General Albert Pike, ca. A.D 1870.

Tom said...

Another potential link of interest: First Things has an article up on Duns Scotus and the univocity of being.

Mr. Green said...

Scott: And surely mathematical statements explain some things.

I took Sean to mean something along the lines of mathematics (though it may explain things about mathematical objects) not explaining anything about physical objects — at least not by itself. A mathematical model of something physical will explain it [partially], but of course the model has to be about something physical.

That's fine as far as it goes, although I'm not certain that intentionality is required: I can model taking two apples and adding two more apples using abstract arithmetic; but can't I also simply add the apples themselves? A model can represent one thing by something different, but the twoness of a pair of apples is right there in the apples… it's not clear to me that a model in some indirect sense is needed. But are units intentional per se? Or is the intentionality so obvious I'm taking it for granted?

Timocrates said...

On the Rorate Caeli Article (Theological: Christian).

Heh. Rorate Caeli can't take heat. They blocked me for criticizing some of their more virulent anti-Islamic sentiments.

Anyways, on the topic linked to in Rorate Caeli, Christ did not preach 'doctrinalism'.

"thou shalt call his name JESUS. For he shall save his people from their sins".

Jesus means salvation, which is in the first place an act. The Lord's first priority is saving humanity from sin. Doctrine is important only relative to that end. That's 'manualism' in the worst sense. It reduces Christ to a philosopher. Now of course there is nothing wrong with being a philosopher or teacher; however, doctrine does not itself accomplish salvation. No mere man could accomplish salvation. Aristotle's doctrine might bring philosophical enlightenment and wisdom but it does not save from sin. "Manualism" - when it is rightly being criticized and not caricatured on being abused for some other reason - is the error of making Christ first a teacher then, almost secondarily, a saviour. No. Christ was a saviour first and foremost and hence his own consistent emphasis on the Cross and the Way of/to the Cross. The Cross - salvation - was what he had to even "must" do.

John West said...

Mr. Green,

A model can represent one thing by something different, but the twoness of a pair of apples is right there in the apples… it's not clear to me that a model in some indirect sense is needed. But are units intentional per se? Or is the intentionality so obvious I'm taking it for granted?

One would think twoness is either instantiated in the pair of apples, or not. If twoness is instantiated, then it is there (completely apart from human mental activity) to be discovered.

seanrobsville said...

@ Mr Green

"I took Sean to mean something along the lines of mathematics (though it may explain things about mathematical objects) not explaining anything about physical objects — at least not by itself. A mathematical model of something physical will explain it [partially], but of course the model has to be about something physical".

Yes, that's what I meant. And someone has to judge which mathematical model is applicable ('about') the system.

Consider the case of the Quality Assurance department in a furniture factory. They are getting lots of complaints about the uneven lengths of the legs on household chairs, but very few about milking stools.

Some members of the team believe the relative frequencies of complaints are a result of the relative stoicism of farming folk compared with city dwellers, and set out to devise a stoicism index to compare rural versus urban populations.

Another group believes that there is a physical explanation, and the difference lies in soil mechanics. Milking stools will tend to settle into a stable configuration due to differential forces of the uneven legs on the compressable soil, whereas no settling is possible on a hard floor.

Then someone like Scott comes along and applies Euclidean geometry...

swordfishtrombone said...

"Has mathematics misled modern science?"

When the title of an article is a question, you can pretty much guarantee that the answer is going to be "no". Brian Appleyard has no clue what he's writing about.

Glenn said...

Consider the case of the Quality Assurance department in a furniture factory. They are getting lots of complaints about the uneven lengths of the legs on household chairs, but very few about milking stools.

Some members of the team believe the relative frequencies of complaints are a result of the relative stoicism of farming folk compared with city dwellers, and set out to devise a stoicism index to compare rural versus urban populations.

Another group believes that there is a physical explanation, and the difference lies in soil mechanics. Milking stools will tend to settle into a stable configuration due to differential forces of the uneven legs on the compressible soil, whereas no settling is possible on a hard floor.

Then someone like Scott comes along and applies Euclidean geometry...


Yeah, well, that's just great.

The company produces both household chairs and milking stools with uneven legs, and there's nothing to worry about.

There's nothing to worry about because milking stools are more likely to be used on compressible soil than on a hard floor, and the stools will tend settle into a stable configuration when used on said soil.

Since the milking stools with uneven legs will tend to settle into stable configuration when used on compressible soil, the Quality Assurance team gives the stools its company's equivalent of the Good Housekeeping Seal of Approval.

But then the company president hears about all this, and calls the Quality Assurance team on the carpet. There are some things he wants to know, such as:

1. Why does one part of the Quality Assurance department think the attitudinal disposition of the company's customers is its proper concern rather than whether the company is producing quality products?

2. Why does another part of the Quality Assurance department think that coming up with a rational explanation for why one set of customers is less likely than another to complain about a crappy product somehow manages to 'uncrapify' that product?

3. Why is Scott the only member of the Quality Assurance department whose concern actually centers on the company's products, and how come he alone tried to address both the household chairs and milking stools, while the rest of the team cared only, and only tangentially, about the milking stools and not at all about the household chairs?

No good answers are forthcoming. But the company president knows what to do nonetheless, so he:

a) reassigns the employees in 1. above to where they belong, i.e., to the marketing department;

b) reassigns the employees in 2. above to where they belong, i.e., the sales department; and,

c) promotes Scott to Executive Vice President of Quality Assurance, and gives him carte blanche to hire a new batch of employees for the Quality Assurance department, each of whom is to understand that Quality Assurance has something to do with assuring the quality of the company's products.

Mr. Green said...

SwordfishTrombone: When the title of an article is a question, you can pretty much guarantee that the answer is going to be "no".


Betteridge's Law strikes again. I don't see what the big deal is, myself. A little deal, maybe — figuring out where to draw the line between the parts of a model that describe reality and the parts that are merely artifacts of the modelling is important, but that's nothing new to science. String-theory is hardly unempirical... if it predicted, say, the surface temperature of the sun to be a comfy 70°F and physicists wanted to stick with it, well, then I could see the problem. But hypothesising that parts of the model we cannot test empirically are nevertheless just as "real" as the rest is part of how scientists work in practice. (Haven't these guys ever seen a periodic table of the elements?) It's somewhat assuring to see interest in philosophical and metaphysical contributions, though. I hope.

Glenn said...

Daniel wrote (under an earlier OP),

There's a book about Aristotle and Actualism I want to bring up in the next 'Links of interest' combox.

Hint, hint.

Scott said...

@Mr. Green:

"A mathematical model of something physical will explain it [partially], but of course the model has to be about something physical."

Sure, and in general a model has to be about something in order to qualify as a model at all. What would make it a "model" otherwise?

I think part of the confusion here is about the word "intrinsically." It's true that the words, symbols, sounds, and so forth used in statements and models, and the statements and models themselves, lack "intrinsic" intentionality. But if that's all that it means to say that mathematical (or any other) statements or models aren't "intrinsically about" anything, then it's irrelevant to seanrobsville's conclusion (that mathematical statements and models don't explain anything).

To reach that conclusion he needs it to mean that mathematical statements and models don't, by nature, have intentional objects at all. And that's absurd: any statement, qua statement, is most certainly "about" something, by its nature as a statement, even if its intentionality is derived* and even if its intentional object is something that doesn't exist (Sherlock Holmes, say).

Likewise a model. A model of a given real-world phenomenon may be good or bad, or one such model may be better or worse than another. But to talk of a "model" at all is already to construe it as a model of something. Even a system of differential equations construed as differential equations isn't a "model" until someone treats it as one.

And here again, the conclusion that a model doesn't "explain" merely because even the mathematics itself, qua mathematics, doesn't tell us what real-world phenomena it might be used to "model" is simply a non sequitur. Indeed, in some instances what makes one "model" better than another is precisely that it has greater explanatory power.

In short, so far as I can see, seanrobsville's claim is either sweeping but wrong, or true but trivial.

----

*Nor would I necessarily agree that the intentionality of a "statement" is derived; that depends on just what we mean by "statement." If we mean the real event of my saying at some particular moment that, say, "2 + 2 = 4," that is, as a deliberate act of stating something, then of course such a statement is intrinsically intentional because the agent making it is.

seanrobsville said...

Does a string of random numbers possess intentionality?

Does it suddenly gain intentionality when Alan Turing decodes it as a message to Nazi U-boats?

Did imaginary and complex numbers possess intentionality during the centuries when they were regarded as mathematical absurdities?

Did they gain intentionality when electrical engineers found they were useful for modelling AC currents?

Glenn said...

seanrobsville,

Does a string of random numbers possess intentionality?

Does it suddenly gain intentionality when Alan Turing decodes it as a message to Nazi U-boats?


Reduce and simplify: Might an encoded message to a boat lack intentionality unless and until it is decoded?

Did imaginary and complex numbers possess intentionality during the centuries when they were regarded as mathematical absurdities?

Did they gain intentionality when electrical engineers found they were useful for modelling AC currents?


Reduce and simplify: Is intentionality lacking when something is known without that something also being known to be useful?

taylormweaver said...

Because this is a "Links of Interest" post, perhaps it is acceptable for me to post a link that brings up questions regarding "neuroscience" and "gender identity"?

I am struggling with this a bit, but this recent HuffingtonPost article claims that there is definite proof that counts definitively against a more strict binary gender construct.

The claim is specifically that now there can be a neurological distinction made between gender identity and biological sex.

Obviously, there are problems with the claims, as it doesn't seem that all oft he data that is being relied upon takes into account hormone therapy. And, so, there may be a correlation problem that isn't really being taken into account. Besides that I don't have the ability to really parse this from a Thomistic point of view.

Just curious if anyone would like to provide any sort of commentary. I don't really have anywhere else to go to discuss this with capable philosophical types, especially capable Thomists.

http://www.huffingtonpost.com/ravishly/neuroscience-proves-what-_b_6494820.html

Daniel said...

@Glenn,

Hint taken! It might be a little anti-climactic though - a few weeks ago I was looking at Zev Bechler's Aristotle and Actuality which bills itself as a full-scale attack on Aristotle’s ‘actualistic ontology’ and wondered if Ed or Oderberg had ever commented on it anywhere. Given that scholastic accounts of Actualism differ probably quite a lot from that of ‘pure Aristotelianism’ due to their endorsement of Divine Exemplarism I don’t know to what extent some of the points would be applicable. Lloyd Gerson absolutely trashes the book in a review available online.

Scott said...

@seanrobsville:

"Does a string of random numbers possess intentionality?

Does it suddenly gain intentionality when Alan Turing decodes it as a message to Nazi U-boats?"

More to the point of your original conclusion:

Does a statement about a string of random numbers have intentionality? And when that statement comes from Alan Turing and he's saying he's decoded it as a message to Nazi U-boats, does it have the power to explain anything?

Dan Franklin said...

Hello Dr. Feser and all other Thomists on this page.
A while back I was talking to a friend of mine who’s an agnostic of sorts. We were talking about the behavior of the subatomic world. He had read in a book by a physicist that if a person were to continuously attempt to walk through a wall that there is a small chance that the behavior of the subatomic particles would permit this to happen: that a person could easily walk through a wall. The chances are slim, but it could still happen. That anything could actually happen. Things that appear solid could, at the whim of the subatomic world, immediately and on their own rearrange themselves.
This view of reality eventually just seemed to be common wisdom. It was kind of upsetting, because it seemed to undermine all constancy of reality…. As well as reinforcing the view that everything is reducible to the smallest, randomly-acting particle that constitutes all there is.

But then I read your book. And for the first time came across notions of things having natures. A nature imposed from above on the constituent matter of below (prime matter). As well as the reasoning for this view.

The reductive view of reality, that the foremost reality is random particles, bothered me for two reasons: 1) it seemed to undercut all meaning to life, solidity and constancy just a flawed interpretation of what we see. And 2) it truly didn’t seem to account for what I saw. I could see the constancy, I couldn’t see the erratic behavior. Now I get the whole “well, you might have to wait millions of years until you see the wackiness”…. But, kind of an underwhelming response. I felt like I was just accepting it because really smart were advocating it.

Anyway, your book and the Thomistic view on nature, essences, forms, substance, prime matter (and the rest of the story) accounts for a reality that I actually do see. As well as accounts for a reality that I have an innate desire for.
Reality may be composed by the seemingly random, uncaring bits of prime matter….. but what we see is the intentional, purposive nature that dictates to the matter what the form is going to be, not the matter capriciously taking on this form opposed to that form.

Dan Franklin said...

Hi Taylormweaver,

I've thought about this before.
I've heard the comments that the neurology of a person shows that gender isn't binary.
But then I get confused: how do they know for certain that what they're viewing is certainly telling them that?
Okay, so this area on a man's brain lights up in a way similar to a woman's.... so?

How can we be certain that the neurological activity or behavior that they're witnessing and mapping actually tells us anything about a person's true biological sex?


Scott said...

@Dan Franklin:

"How can we be certain that the neurological activity or behavior that they're witnessing and mapping actually tells us anything about a person's true biological sex?"

They don't say we can. Biological sex is binary, and in normal cases nobody needs magnetic resonance imaging to tell a biologically male human from a biologically female one. This is about "gender identity," which is another matter entirely and a much slipperier one.

And I think that's what you probably meant to ask about: how do they know these differences in brain chemistry indicate anything about anyone's "true gender"?

And that's a very good question. I don't see anything surprising or unexpected (or for that matter contrary to Thomism; far from it, in fact) in the results themselves, but I haven't seen any reason to think they correlate with anyone's "true gender." In order to establish such a correlation, we'd have to have some other way to tell what someone's "true gender" is, and not only do I not know of any, I'm not even persuaded the idea is meaningful.

For that matter, even if the results did establish such a correlation, there still wouldn't be anything normative about them. Anyone who thinks there's something disordered about a biological male's wanting to be female could simply regard this as a possible basis for a diagnostic tool.

Nor would such a correlation establish causation. It wouldn't, for example, support the conclusion that people have the gender identities they have because their white matter behaves thus-and-so, any more than other neurological research supports the conclusion that I choose to move my hand because certain neurons are firing in certain ways.

In short, it's somewhat interesting research, but it's very likely to be cited in support of moral implications that it does not in fact have.

Scott said...

(That is, I'm not persuaded that the idea of "true gender" is meaningful if it's distinct from one's biological sex.)

Glenn said...

Daniel,

Danke.

- - - - -

Regarding the earlier matter of my setting my sights low (by looking forward only to what the book might be rather than to the book itself), it looks like I get off scot-free (this time, anyway):

"[Bechler's Aristotle's Theory of Actuality] is not a work whose imaginative hypotheses are much constricted by fidelity to the texts." -- Lloyd Gerson (Here.)

Matt Sheean said...

Scott and weaver,

"we'd have to have some other way to tell what someone's "true gender" is"

As I've encountered this as expressed "in support of moral implications that it does not in fact have" the "other way" is that gender that the person reports experiencing themselves as. Of course, all this shows it that there is a neurological basis for the experience, but that, as Scott notes, is hardly surprising!

There are other such distinctions to be made between subjective identity and what is objectively available to everyone else with respect to the neurological, and in most every other case when the two don't line up we take that to mean something is amiss.

Scott's more or less said this, but I wanted to expound a bit on the way this information is (mis)used. It's strange that it's used at all since the whole notion of a gender spectrum and such is more fitted to some (usually continental) species of anti-realism (more specifically, some kind of constructivism) that, for philosophical reasons, could just as easily be indifferent to the science.

John West said...

Tom writes: Another potential link of interest: First Things has an article up on Duns Scotus and the univocity of being.

Does anyone know of any longer, Thomist treatment of Scotist concerns about analogy?

Irish Thomist said...

@Dan Franklin & Scott etc.

In the case of complete androgen insensitivity syndrome (CAIS) one might consider them to be female by different genetic means since a proper understanding of what the Y does and the X does in terms of genetic material and information is important to this consideration. One who does not understand genetics might jump to an inaccurate conclusion here.

Scott said...

@Irish Thomist:

True, but not exactly a "normal case." As I understand it, those with CAIS are phenotypically female (and ordinarily also have female "gender identities") because their genetic masculinity can't be expressed or manifested. With respect to the normal biological function of genes, this is a privation.

John West said...

Enjoying your new promotion, I see.

Scott said...

@John West:

Heck, yeah. You are looking at the new Executive Vice President of Quality Assurance. And I'm here to assure you that things have qualities.

Irish Thomist said...

@Scott

True it's not a normal case but worth bringing up I think you will agree. It's also not the only (although maybe the best) example.

Scott said...

@Irish Thomist:

Agreed on all counts. I just wanted to make clear that it's not a counterexample to my own careful statement that in normal cases nobody needs magnetic resonance imaging to tell a biologically male human from a biologically female one.

One thing it definitely is is a good example of why someone might have a "gender identity" that didn't match his/her sex genotype.

Irish Thomist said...

@Scott

Agreed.

Note: Off topic
BTW did you get a chance to have a look at the list of suggested books to read for the era mentioned on my blog? I'm sure you would have some additional suggestions.

Irish Thomist said...

@Scott

Never mind. I see you commented on the 20th on that Post! Will have to work on the next list.

Kevin Jessup said...

Here's an old post from Martin Gardner (former Scientific American columnist) on the subject. I always liked this one...

http://www.csicop.org/si/show/multiverses_and_blackberries/

Scott said...

@Kevin Jessup:

"Here's an old post from Martin Gardner (former Scientific American columnist) on the subject."

Good post, but on what subject?

Anonymous said...

What do the people here think of this argument?

(1) All concrete objects that have an originating or sustaining cause have a material cause of their existence.
(2) If classical theism is true, then the universe is a concrete object that has an originating or sustaining cause without a material cause of its existence.
(3) Therefore, classical theism is false.

(1) is supported by arguments parallel to those used to support the principle of causality.

The argument is from here: http://exapologist.blogspot.com/2014/12/quick-sketch-of-case-against-classical.html

Scott said...

@Anonymous:

We've seen this argument before, but I'll try not to yawn.

Speaking only for myself, I'd say the main thing wrong with it is that the first premise is false in the sense required by the argument: it's simply not the case that every concrete object has a material cause that precedes it in time. If God creates a bronze statue ex nihilo, the statue still has a material cause in the Aristotelian sense, namely the bronze.

I'm sure Aristotle thought things were generally made out of pre-existing materials, but nothing in the concept of a material cause requires that.

John West said...

Scott,

Also, isn't the Scholastic use of material much broader?

Anonymous said...

Scott:

What do you think of the arguments presented in favor of (1)?

John West said...

I suppose I should clarify. My point was that the argument seemed to me to imply a physicalist view of matter.

Anonymous said...

Scott:

The arguments for (1) are found here: http://exapologist.blogspot.com/2014/12/theism-and-material-causality.html

"suppose we were told that a certain log cabin had the following special characteristic: it popped into existence out of nothing without any cause whatsoever. Most, I imagine – including most of those who have read their Hume -- would find such a claim strongly counterintuitive, if not absurd. But suppose instead we were told the cabin was special for another reason: a lumberjack created it without any materials whatsoever. I imagine most would likewise find such a claim absurd or strongly counterintuitive."

"our uniform experience is such that whenever we find a concrete object with an originating or sustaining cause, we also find it to have a material cause. Furthermore, there seem to be no clear counterexamples to PMC (principle of material causality) in our experience. What explains this? PMC is a simple hypothesis, which, if true, would best explain this data. Experience thus provides significant abductive support for PMC."

Glenn said...

John,

I suppose I should clarify. My point was that the argument seemed to me to imply a physicalist view of matter.

Of course it does. And since CT doesn't hold that only proximate matter can be a material cause, the argument either is based on a misunderstanding or is the fruit of willful blindess.

Scott said...

"What do you think of the arguments presented in favor of (1)?"

What do you think of this argument? "Our uniform experience is such that whenever we encounter a personal being, that being has a material body. If we were told that there was a personal being without a material body, most of us would find such a claim absurd or strongly counterintuitive."

Scott said...

@John West:

"Also, isn't the Scholastic use of material much broader?"

Yes, and I agree with Glenn.

Anonymous said...

I think your argument has some defeasible force. What's your point?

John West said...

Glenn and Scott,

Thank you.

Since someone else has brought it up, I've been meaning to ask. Has anyone written any type of analysis of how God might cause worldly happenings?

Daniel said...

suppose we were told that a certain log cabin had the following special characteristic: it popped into existence out of nothing without any cause whatsoever. Most, I imagine – including most of those who have read their Hume -- would find such a claim strongly counter-intuitive, if not absurd. But suppose instead we were told the cabin was special for another reason: a lumberjack created it without any materials whatsoever. I imagine most would likewise find such a claim absurd or strongly counter-intuitive.

From this all the person in question can claim is that both instances outlined sound 'counter-intuitive'. The trouble is the Classical Theist does not base his case against A on the basis of A's just sounding weird but on other claims e.g. that the implicit denial of the PSR therein and admission of Brute Facts leads to incoherency. Unless the critic of B can give other arguments they’re stuck with just saying 'gee that sounds weird'*.

*Thomas, more than so than is necessary I think, would admit that it sounds odd to us our intellect's being geared primarily to the material things of this world.

Also the claim that there is no clear counter-example to the ‘PMC’ is question-begging. The Classical Theist qua proponent of a Cosmological Argument for instance will just turn round and say 'we've just demonstrated the existence of one to you'. Even if God were the only immaterial being it would have no force. Of course as it happens one could also proceed to show that, for instance, the human mind/soul is also an immaterial being and we have no shortage of them!

Scott said...

"What's your point?"

That metaphysical arguments trump vague and uncertain generalizations from our experience with medium-sized dry goods.

John West said...

Daniel,

Even if God were the only immaterial being it would have no force. Of course as it happens one could also proceed to show that, for instance, the human mind/soul is also an immaterial being and we have no shortage of them!

In the latter case, even appeals to weirdness would be dubious. For example, Kripke's modal argument gives good reason to think an immaterial mind is quite intuitive. Descartes, of course, would have had even more to say about it.

Anonymous said...

"That metaphysical arguments trump vague and uncertain generalizations from our experience with medium-sized dry goods."

But this point also applies to the parallel argument from the PSR: "Our uniform experience is such that whenever we encounter a contingent being, that being has a cause of its existence. If we were told that there was a contingent being without a cause, most of us would find such a claim absurd or strongly counterintuitive."

Also, are you suggesting empirical generalizations carry no evidential weight?

Lastly, what about the argument for PMC via abductive inference - PMC is the simplest explanation for our experience? This hasn't been commented on.

John West said...

Anonymous,

But this point also applies to the parallel argument from the PSR: "Our uniform experience is such that whenever we encounter a contingent being, that being has a cause of its existence. If we were told that there was a contingent being without a cause, most of us would find such a claim absurd or strongly counterintuitive."

One problem with your PMC isn't that the cabin analogy is totally, completely, and utterly useless; it's that the analogy is trumped by metaphysical arguments, and that you don't offer much else. By comparison, the PSR has other arguments for it and no effective arguments against it.

John West said...

Also, I dislike your use of absurd. As I see it, absurd is used to connotate straightforward, logical contradictions as in reductio ad absurdum; you're using it in the sense of "very strange".

Anonymous said...

John West:

To be clear, I'm just relaying the arguments from here: http://exapologist.blogspot.com/2014/12/theism-and-material-causality.html

Besides the appeal to intuition, there's also the argument to PMC via abductive inference - PMC is the simplest explanation for our experience.

Anonymous said...

Re-posted a knock down argument against classical theism. What do you people think?

1. Time exists only as part of space-time.
2. Time does not exist, therefore, outside of our universe.
3. To create the universe one must necessarily be "outside" of it.
4. If god exists outside of the universe, there is no time.
5. Without time there can be no change - there cannot be one temporal "moment" where the universe does not exist, and another where it does.
6. Therefore god cannot have created the universe.

Daniel said...

Also, are you suggesting empirical generalizations carry no evidential weight?

Let's imagine we challenge Hume on these grounds. Even without reference to the famous Problem of Induction he would be happy to accept that it sounded absolutely bizarre to us for a being to come into existence without a cause and that perhaps no being had ever yet done so, however his argument had shown that it was still logically possible for such a thing to happen. To challenge him one must challenge the reasoning behind the argument (for instance the thesis that conceivability entails possibility or that what was being concived was too general to warrant his conclusion - who knows perhaps what he was conceiving was really an instance of creato ex nihilo). The same holds for case B

Lastly, what about the argument for PMC via abductive inference - PMC is the simplest explanation for our experience? This hasn't been commented on.

A supposed inference to best explanation is vacuous if one can provide non-inferential evidence to the contrary of one of its hypothetical claims i.e. that we know already that immaterial beings exist. If one wants to employ such an argument a la Smart one has to show that all prexisting arguments for immaterial substances, both God and souls, fail for independent reasons. Then one could claim that it seems unlikely that immaterial substances exist.

Glenn said...

Anonymous,

Re-posted a knock down argument against classical theism. What do you people think?

1. Time exists only as part of space-time.
2. Time does not exist, therefore, outside of our universe.
3. To create the universe one must necessarily be "outside" of it.
4. If god exists outside of the universe, there is no time.
5. Without time there can be no change - there cannot be one temporal "moment" where the universe does not exist, and another where it does.
6. Therefore god cannot have created the universe. 1. Time exists only as part of space-time.


Surely, sense is made when, given premises 1 and 2, it is said that if God exists outside our universe then time does not exist in God.

But how, given the same two premises, does premise 4 make sense? That is, how does it follow from premises 1 and 2 that God existing outside our universe entails time not existing inside our universe?

Also, and at least according to St. Thomas, creation is not change, so it is somewhat less than clear why it might be thought that the CT claim of God having created the universe can be successfully argued against by talking about the lack of change in the absence of time.

Glenn said...

(s/b "...it is, from a Thomistic perspective, somewhat less than clear why...")

Glenn said...

Anonymous,

Lastly, what about the argument for PMC via abductive inference - PMC is the simplest explanation for our experience? This hasn't been commented on.

exapologist observed that immaterial things -- such as shapes, surfaces, events, propositions, numbers, sets, etc. -- do exist, and, apparently, hypothesized that his argument might have greater traction were he to define "concrete object" so as to exclude any and all concrete entities which are immaterial (such as a shapes, surfaces, events...).

Glenn said...

Here's a knock down proof against the silly, nay, absurd notion that coins are money:

1. Only a monetary object counts as money.

2. I defined a monetary object as any monetary entity made of paper.

3. Coins are monetary entities.

4. But coins are not made of paper, so coins are not monetary objects.

5. Ergo, coins are not money.

Daniel said...

Quick remarks:

The argument he gives obviously only applies to temporal creation i.e. the universe's having a temporal beginning. In that sense and minus the modern nonsense about space-time Pre-Christian Classical Theists may well have thought along similar lines.

Point 5 is strange though since most scholastic metaphysics of temporal creation from Augustine onwards depend upon that very point.

The whole thing revolves around ridiculously simplistic understandings of what constitutes Time though.

Anonymous said...

About the earlier "knock down" argument, time was conceived as per Hawking's writings:

The problem of what happens at the beginning of time is a bit like the question of what happened at the edge of the world, when people thought the world was flat. Is the world a flat plate with the sea pouring over the edge? I have tested this experimentally. I have been round the world, and I have not fallen off. As we all know, the problem of what happens at the edge of the world was solved when people realized that the world was not a flat plate, but a curved surface. Time however, seemed to be different. It appeared to be separate from space, and to be like a model railway track. If it had a beginning, there would have to be someone to set the trains going. Einstein's General Theory of Relativity unified time and space as spacetime, but time was still different from space and was like a corridor, which either had a beginning and end, or went on forever. However, when one combines General Relativity with Quantum Theory, Jim Hartle and I realized that time can behave like another direction in space under extreme conditions. This means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world.

Time

Glenn said...

Anonymous,

About the earlier "knock down" argument, time was conceived as per Hawking's writings:

...This means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world.


Well, the "knock down" argument claims that time, conceived as per Hawkins' writings, cannot exist outside our universe.

But Hawkings has also claimed that the universe can and will create itself from nothing.

Now, if our universe did indeed create itself from nothing, then -- and aside from the fact in this case our universe was created (though by Whom or what we need not inquire into just now) -- it follows that there was nothing before our universe.

But if there was nothing before our universe, then time, which the knock down argument says cannot exist outside our universe, did not exist before our universe.

It would appear, then, that is a slight conflict between Hawkins' conception of time, and the knock down argument which draws on that conception.

Glenn said...

(s/b "Now, if indeed our universe was created from nothing...")

Anonymous said...

So guys is this a knock down or not? Can we abandon Thomism over this piece of brilliance?

John West said...

Anonymous,

I did think perhaps you were this ex-apologist fellow. Sorry for the mistake.

Besides the appeal to intuition, there's also the argument to PMC via abductive inference - PMC is the simplest explanation for our experience.

Clearly it's not the best explanation if it conflicts with the conclusions of other, successful metaphysical arguments.

John West said...

So guys is this a knock down or not? Can we abandon Thomism over this piece of brilliance?

I think I'll hold off and see what else they have.

John West said...

More grist for the mill:

The argument is simple: We've observed a huge quantity of an extremely wide variety of concrete objects, and all of the concrete objects we've observed are contingent; so, probably, all concrete objects whatsoever are contingent. But no Anselmian being is contingent. So, probably, there are no Anselmian beings. (A Quick Inductive Argument Against Anselmian Beings.

Daniel said...

Does anyone else get the impression that theists spend more time reading theist philosophers of religion than atheists do reading atheist philosophers of the same discipline?

On a different note I am not sure Christianity has lost much with this fellow...

Scott said...

"But this point also applies to the parallel argument from the PSR: 'Our uniform experience is such that whenever we encounter a contingent being, that being has a cause of its existence. If we were told that there was a contingent being without a cause, most of us would find such a claim absurd or strongly counterintuitive.'"

And in this case the metaphysical argument would support the empirical generalization, not undermine it.

"Also, are you suggesting empirical generalizations carry no evidential weight?"

I don't quite know how anyone might manage to read that out of what I wrote.

"Lastly, what about the argument for PMC via abductive inference - PMC is the simplest explanation for our experience?"

What about it? As John West says, it's not the best explanation if it can be shown on metaphysical grounds to be false.

John West said...

Daniel, Scott, Glenn, Anonymous, etc.,

I do have a quibble with Thomism (or my probably still lacking understanding of it) and “formulas--equations...” though. I'm worried Scholastic realism, in tandem with the doctrine of analogy and Divine Simplicity, is implicitly anti-realist about mathematics.

One either knows a mathematical truth, or does not. But it seems to me that if a mathematical truth is grounded in and reflects an aspect of the divine essence, then if we know it we have knowledge of God. Not of the entirety of God — that's impossible — but of the aspect[1] of God reflected under certain constraints. To say that the mathematical truth we know is only an analogous understanding of the true, deeper nature it reflects is (it seems to me) simply to concede anti-realism about mathematics by saying mathematics is a symbol of something different (or for describing something else) — that's the same thing anti-realists say.

[1]Construe an “aspect of x” in the sense of “a way x can be perceived by the mind.”

Glenn said...

John,

Interesting quibble.

Hm.

If there is agreement that we cannot know God in His entirety, then there seems to be agreement that we cannot know Him perfectly. And if there is agreement that we cannot know Him perfectly, then there seems to be agreement that we can God only imperfectly. But if there is agreement that we can know God only imperfectly, whence the quibble that any particular type of knowledge enables us to know Him no better than analogously?

Since that may not qualm the quibble, let me ask this:

Given that all being is from God, is to know a being who/which is not God to know God Himself?

Glenn said...

That's not meant to be a 'thought stopping' question.

Suppose the answer is, "If the being known is a mathematical being, then, yes, to know a being which is not God is to know God Himself (even if only imperfectly)."

What might the next question be?

That is, what additional question(s) might be suggested by the answer supposed?

Glenn said...

That's not meant to be a 'thought stopping' question.

Perhaps not. But making reference to a question that hasn't been asked surely is.

I don't know if I somehow forgot to post the comment I meant to continue, or the comment was posted but just hasn't shown up yet.

Repeat:

John,

Interesting quibble.

Hm.

If there is agreement that we cannot know God in His entirety, then there seems to be agreement that we cannot know Him perfectly. But if there is agreement that we cannot know Him perfectly, then there seems to be agreement that we can God only imperfectly. And if there is agreement that we can know God only imperfectly, whence the quibble that any particular type of knowledge enables us to know Him no better than analogously?

Since that may not qualm the quibble, let me ask this:

Given that all being is from God, is to know a being who/which is not God to know God Himself?

Scott said...

@John West:

The physical universe is (so we believe) finite, so suppose we choose a number larger than any that is physically instantiated. Either that number is prime or it isn't. So there's at least one mathematical fact that doesn't depend for its truth on physical instantiation.

I'm not prepared, in Augustine's phrase, to leap headlong into the pit of impiety and declare that God knows not all numbers; I think mathematical truths are about something whether or not that something is physically instantiated. If it turns out that Aquinas disagrees, then I'll just have to think he was mistaken.

Scott said...

I don't think divine simplicity is thereby endangered, however. Any account of divine simplicity allows that God can know more than one thing; any multiplicity is in the objects of the divine intellect, not in the divine intellect itself. Assuming that the doctrine of divine simplicity makes sense in the first place (which I think it does), I don't see that mathematical truths pose any more of a problem for it than any other truths.

John West said...

Glenn,

Thanks for the reply.

If there is agreement that we cannot know God in His entirety, then there seems to be agreement that we cannot know Him perfectly.

I agree. But I would just reply that to know an aspect of God non-analogously is not to know Him perfectly. The knowledge, however, isn't imperfect because it's analogous. It's imperfect because it's only an aspect of God and not the entirety of God. If that still counts as analogous, then it may be I just have a crude understanding of “analogy”.

Given that all being is from God, is to know a being who/which is not God to know God Himself?

The difference is that there's something for statements about beings to pick out. There's nothing for statements about mathematics not instantiated in the physical part of the world to pick out.

In other words, my concern is that mathematical statements would be vacuous; or, if they simply pick out God, then on (my hopefully correct understanding of) the Thomist view they pick out something to which mathematics is only analogous. But this is what a lot of anti-realists would say—that statements about mathematics pick out something non-mathematical, and mathematics is just a modeling language we use for it*.

Given that all being is from God, is to know a being who/which is not God to know God Himself?

Just so it doesn't seem I'm being somehow disingenuous and dodging your question: “Yes.” I would say it's to have knowledge of some aspect of the Divine Essence.


*Though some further quibbling over "close enough" may be possible here.
**I'm not grounding mathematics in the Divine Mind here, because it seems to me that God could have had different thoughts. I think that unless somehow ultimately grounded in the divine essence or nature, that would make necessary mathematical truths subject to and contingent on divine fiat. But surely necessary truths aren't contingent.

John West said...

Scott,

I don't think divine simplicity is thereby endangered, however. Any account of divine simplicity allows that God can know more than one thing; any multiplicity is in the objects of the divine intellect, not in the divine intellect itself. Assuming that the doctrine of divine simplicity makes sense in the first place (which I think it does), I don't see that mathematical truths pose any more of a problem for it than any other truths

I agree. I don't think any of this endangers divine simplicity at all.

Well, except possibly in the political sense of it opening the door for others taking it too far (ie. I think I read that some historians have complained Scotus opened the door for the rejection of simplicity). But either way, I don't we should make philosophical rulings on the basis of political usefulness or otherwise.

John West said...

But I would just reply that to know an aspect of God non-analogously is not to know Him perfectly. The knowledge, however, isn't imperfect because it's analogous. It's imperfect because it's only an aspect of God and not the entirety of God.

A light analogy because I'm still worried I'm writing unclearly. Say someone is showing me a completely smooth, round gem, but that someone blocks all but a dime-sized amount of it from my sight. No one would say seeing that not-blocked amount does not give me some knowledge of the gem. I just wouldn't have all knowledge there is to have about the gem.

John West said...

The second last sentence should read: "[...] not-blocked amount does not give me some knowledge *about* the gem. [...]"^

Glenn said...

John,

I'm still worried I'm writing unclearly.

I too have calms I might not be writing clearly. ("Since that may not qualm the quibble...")

Glenn said...

John,

But I would just reply that to know an aspect of God non-analogously is not to know Him perfectly. The knowledge, however, isn't imperfect because it's analogous. It's imperfect because it's only an aspect of God and not the entirety of God.

I follow the last two sentences, but am not so sure I follow the first sentence. If by 'non-analogously' you mean without the mediation of, if I may broaden the referent, analogy, figure, sign, symbol, etc., then I must say I don't know in what way that might be possible. The only alternative would seem to be to know an aspect of God directly. But if St. Thomas is correct that we cannot know even the essence of our own intellects directly, then I don't see how we might know even a small apect of God's essence directly, with or without the aid of mathematical objects (or anything else).

Glenn said...

John,

Turning to another Scholastic, Nicolas of Cusa... Jasper Hopkins writes the following in note 12 of his introduction to his translation of Cusa's Learned Ignorance):

In DP 43 the Abbot (John Andrea) protests that our knowledge of mathematical truths is exact knowledge. Nicholas does not deny this but instead makes the following distinction: "Regarding mathematical [entities], which proceed from our reason and which we experience to be in us as in their source [principium]: they are known by us as our entities and as rational entities; [and they are known] precisely, by our reason's precision, from which they proceed. . . . Without these notional entities reason could not proceed with its work, e.g., with building, measuring, and so on. But the divine works, which proceed from the divine intellect, remain unknown to us precisely as they are. If we know something about them, we surmise it by likening a figure to a form. Hence there is no precise knowledge of any of God's works, except on the part of God, who does all these works. If we have any knowledge of them, we derive it from the symbolism and the mirror of [our] mathematical knowledge."

(DP is De Posset, and 43 is on PDF page 936.)

Glenn said...

(Oy vey. The 3rd link unnecessarily duplicates the 2nd link.)

Glenn said...

(...and 43 is on PDF page 936.

(Well, 43 starts on page 935, although the quotation is from 936.

((Errors and inaccuracies popping up left and right. Little wonder: now 8pm, and I've been on the computer since 3:30am. Time for a looong break, methinks.))

Daniel said...

What if one were to just deny Analogical predication, at least in some areas, and claim that mathematics pertains directly to God in as much as One is a Transcendental (whether one can treat this as the number 1 or as something more basic, an empty set or something, is another question).

But if St. Thomas is correct that we cannot know even the essence of our own intellects directly

A long time since I checked this but doesn't Thomas claim that, though we do possess apperception and thus are always aware of our thinking, we are only properly aware of ourselves as a thinker in the cogito sense upon reflection?

Glenn said...

Daniel,

What if one were to just deny Analogical predication, at least in some areas, and claim that mathematics pertains directly to God in as much as One is a Transcendental (whether one can treat this as the number 1 or as something more basic, an empty set or something, is another question).

The ease with which the phrase "in as much as" is used to qualify the claim seems to indicate a tacit awareness that "directly to God" is more a manner of speaking than an actual fact.

o [T]he angel knows God in as much as he is the image of God. Yet he does not behold God's essence; because no created likeness is sufficient to represent the Divine essence. ST 1.56.3

o [T]he intellect does not apprehend things according to their mode, but according to its own mode. Hence material things which are below our intellect exist in our intellect in a simpler mode than they exist in themselves. Angelic substances, on the other hand, are above our intellect; and hence our intellect cannot attain to apprehend them, as they are in themselves, but by its own mode, according as it apprehends composite things; and in this way also it apprehends God[.] -- ST 1.50.2

A long time since I checked this but doesn't Thomas claim that, though we do possess apperception and thus are always aware of our thinking, we are only properly aware of ourselves as a thinker in the cogito sense upon reflection?

While it doesn't seem to be quite what you're referring to, he does say that "the intellect knows itself not by its essence, but by its act", and that, "The mind knows itself by means of itself, because at length it acquires knowledge of itself, though led thereto by its own act" (ST 1.87.1 (and ad 1)).

Irish Thomist said...

@Anonymous

Just to be crystal clear that when some physicist's today claim time to now be explained by science then make the bold affirmation that it is either not real or no longer a metaphysical problem they are simply wrong.

Why are they wrong? For the same reason that it was a philosophical matter in the first place really. It's much like neuroscience telling us how the brain works then someone deciding that consciousness does not exist. The conclusion does not follow.

John West said...

Glenn,

Those Nicolas of Cusa quotes were very informative. Thanks. I'll have to take a look at the whole work later.

To your first post:

I follow the last two sentences, but am not so sure I follow the first sentence. If by 'non-analogously' you mean without the mediation of, if I may broaden the referent, analogy, figure, sign, symbol, etc., then I must say I don't know in what way that might be possible. The only alternative would seem to be to know an aspect of God directly. But if St. Thomas is correct that we cannot know even the essence of our own intellects directly, then I don't see how we might know even a small apect of God's essence directly, with or without the aid of mathematical objects (or anything else).

Directly (maybe univocally, if that's appropriate) works.

Assume St. Thomas is correct and we cannot know even the essence of our own intellects directly, and that it follows we can't know even a small aspect of God's Essence directly. Then my original concern:

that [statements about mathematics not instantiated in the physical part of the world] would be vacuous; or, if they simply pick out God, then on (my hopefully correct understanding of) the Thomist view they pick out something to which mathematics is only analogous. But this is what a lot of anti-realists would say—that statements about mathematics pick out something non-mathematical, and mathematics is just a modeling language we use for it*.

seems to have been well-placed, for mathematics-not-physically-instantiated (at least) are merely a modelling language we use for something else (and in this case, that something-else “remain[s] unknown precisely as [it is]” (Cusa)). They're just symbols of something different, like the anti-realists write.

But to state that facts like whether a number larger than any physically instantiated is prime or not, or statements of the form a+b=c*, where a, b, and c are variables, aren't true for sufficiently large, non-physically instantiated numbers (because ultimately vacuous) is a huge deal.


*the same form as 2+2=4.
**I was going to ask: “Also, even if not the divine essence, what about the Divine Ideas (say some of them are ultimately grounded in the Divine Nature, and are thoughts God would have thought no matter what else God could have thought?)” But Nicolas of Cusa seems to be saying even then, no.

John West said...

Daniel and Glenn,

What if one were to just deny Analogical predication, at least in some areas, and claim that mathematics pertains directly to God in as much as One is a Transcendental (whether one can treat this as the number 1 or as something more basic, an empty set or something, is another question).

I read "in as much as One is a Transcendental" simply to mean "in cases such as One, which is a Transcendental". I'll wait until Daniel clarifies before commenting on it.

Daniel said...

@John West,

Yes, that's correct - teaches me for typing in a hurry!

Daniel said...

@Glenn,

I think what I had in mind came from somewhere in On Truth. I know for a fact that Coplestone gives a summary very similar to what I said in his intro to Aquinas though

About the quotes you give: I took them to be just another application of the maxim 'As a thing is so does it act: Action is the index of Essence' in that we do not know a priori that our intellect is immaterial/our soul is an immaterial substance but we can learn in by reasoning from its abilities either in an introspectionist way a la Augustine's argument in On the Trinity or the preferred Aristotelian way re our apprehension of universals. So we can know the essence of our soul but it is not immediately present to us its entirety in the way God enjoys.

Glenn said...

Daniel,

Thanks for the expansion.

Glenn said...

John,

Assume St. Thomas is correct and we cannot know even the essence of our own intellects directly, and that it follows we can't know even a small aspect of God's Essence directly. Then my original concern:

that [statements about mathematics not instantiated in the physical part of the world] would be vacuous; or, if they simply pick out God, then on (my hopefully correct understanding of) the Thomist view they pick out something to which mathematics is only analogous. But this is what a lot of anti-realists would say—that statements about mathematics pick out something non-mathematical, and mathematics is just a modeling language we use for it*.

seems to have been well-placed, for mathematics-not-physically-instantiated (at least) are merely a modelling language we use for something else (and in this case, that something-else “remain[s] unknown precisely as [it is]” (Cusa)). They're just symbols of something different, like the anti-realists write.

But to state that facts like whether a number larger than any physically instantiated is prime or not, or statements of the form a+b=c*, where a, b, and c are variables, aren't true for sufficiently large, non-physically instantiated numbers (because ultimately vacuous) is a huge deal.


I'm not clear on why St. Thomas being correct on that point would relegate mathematics to some shanty-town on the wrong side of the tracks. If St. Thomas is right, then mathematical entities are beings of reason rather than real beings. That something is a being of reason rather than a real being does not mean that statements about it are vacuous, that it is rightly disparaged as being merely this or that, or that it ought to be dismissed or shooed away as just... whatever. There are many things on St. Thomas' account which are beings of reason rather than real beings. He takes them seriously, and has much to say about them. Had he thought that statements about beings of reason were "vacuous", he would have had much less to say than he actually said.

Those Nicolas of Cusa quotes were very informative. Thanks. I'll have to take a look at the whole work later.

Not long ago Daniel had posted a link to Moran's Phenomenology 2010, which has a chapter on Cantor and absolute infinity. Cusa's Learned Ignorance was mentioned in the chapter as possibly having had an influence on Cantor's work, so I tracked it down. That's how I came upon Hopkins' site, his translation of LI, and the quotation given above.

John West said...

Glenn,

Sorry. Could you expand on what a being of reason is for me, or point me to a relevant Stanford article? I had better make sure it's what I think it is before replying.

John West said...

Glenn,

I read the Moran article. Thank you. Though I'm still somewhat uncertain what a being of reason is, I'll try and nail it down with some casework:

I'm not clear on why St. Thomas being correct on that point would relegate mathematics to some shanty-town on the wrong side of the tracks. If St. Thomas is right, then mathematical entities are beings of reason rather than real beings. That something is a being of reason rather than a real being does not mean that statements about it are vacuous, that it is rightly disparaged as being merely this or that, or that it ought to be dismissed or shooed away as just... whatever. There are many things on St. Thomas' account which are beings of reason rather than real beings. He takes them seriously, and has much to say about them. Had he thought that statements about beings of reason were "vacuous", he would have had much less to say than he actually said.

If they're symbols (I mean the numbers, not the numerals) not referring to mathematics as mental objects, then my original concern reappears.[1]

So though I'm not quite sure what you're saying. If you're referring to the idea that our mathematical singular terms refer to mental objects in human minds, then there are a number of problems. One problem is that this view seems incapable of accounting for talk about the class of all real numbers, since human minds could never construct them all. Another problem is that it seems to entail that assertions about very large numbers (in particular, numbers that no one has thought about) are all untrue. For, if no one has ever mentally constructed some very large number, then according to this view any proposition about that number will be vacuous. A third problem with saying that mathematics has being in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.

Presumably you don't mean that mathematical statements refer to Ideas in God's Mind. One problem with that is that it would make necessary mathematical truths contingent on divine fiat, or at very least make mathematical truths contingent (for surely God could have had different thoughts and we don't for example live in the only possible world). It also entails that we can know God's Ideas, which is what Cusa and (if I've understood correctly) you deny.


[1] Similarly, if mathematics (the numbers, not numerals) are symbols of something-else, then I can think of no reason other symbols could not have been used to represent that something-else. But this conflicts with the indispensability thesis.

To be clear, the indispensability thesis doesn't refer specifically to Quine or any of his peculiar methodology. The indispensability thesis is, rather, the thesis at the core of all indispensability arguments about mathematics.

John West said...

Daniel,

Thanks for clarifying.

What if one were to just deny Analogical predication, at least in some areas, and claim that mathematics pertains directly to God in as much as One is a Transcendental (whether one can treat this as the number 1 or as something more basic, an empty set or something, is another question).

And it's not like the view you write about is un-Scholastic or un-classical theist either. Unless given some reason to not be, I'm at least sympathetic to this (or an even more Scotist view.)

John West said...

That center paragraph is supposed to be a quote. Sorry for the deluge of posts. :/

Glenn said...

John,

DEE 4: "We should notice, therefore, that the word 'being,' taken without qualifiers, has two uses, as the Philosopher says in the fifth book of the Metaphysics. (1) In one way, it is used apropos of what is divided into the ten genera; (2) in another way, it is used to signify the truth of propositions. The difference between the two is that in the second way everything about which we can form an affirmative proposition can be called a being, even though it posits nothing in reality."

A being in the first way is a "real being". A being in the second way is a "being of reason".

- - - - -

And on that note, I should say that my earlier assertion -- "if St. Thomas is right, then mathematical entities are beings of reason rather than real beings" -- appears to be: a) to be kind to myself, overly general; b) to be pragmatic, partly true and partly false; and, c) to speak in a manner more my wont, somewhat suspect. From Maurer's Thomists and Thomas Aquinas on the Foundation of Mathematics:

"Some modern thomists, claiming to follow the lead of Thomas Aquinas, hold that the objects of the types of mathematics known in the thirteenth century, such as the arithmetic of whole numbers and Euclidean geometry, are real entities. In scholastic terms they are not beings of reason (entia rationis) but real beings (entia realia). In his once-popular scholastic manual, Elementa Philosophiae Aristotelico-Thomisticae, Joseph Gredt maintains that, according to Aristotle and Thomas Aquinas, the object of mathematics is real quantity, either discrete quantity in arithmetic or continuous quantity in geometry. The mathematician considers the essence of quantity in abstraction from its relation to real existence in bodily substance. 'When quantity is considered in this way,' he writes, 'it is not a being of reason (ens rationis) but a real being (ens reale). Nevertheless it is so abstractly considered that it leaves out of account both real and conceptual existence.'

"Recent mathematicians, Gredt continues, extend their speculation to fictitious quantity, which has conceptual but not real being; for example, the fourth dimension, which by its essence positively excludes a relation to real existence. According to Gredt this is a special, transcendental mathematics essentially distinct from 'real mathematics,' and belonging to it only by reduction.

"Jacques Maritain read the works of Gredt, including his Elementa, and in his magistral Degrees of Knowledge he agrees with Gredt that at least the objects of Euclidean geometry and the arithmetic of whole numbers are entia realia in distinction to the objects of modern types of mathematics, which he calls entia rationis. The objects of the former types of mathematics, Maritain says, are real in the philosophical sense that they can exist outside the mind in the physical world, whereas the objects of the newer types of mathematics cannot so exist. A point, a line, and a whole number are real beings, but not irrational numbers or the constructions of non-Euclidean geometries."

Glenn said...

The ink at this link has yet to fade: Thomists and Thomas Aquinas on the Foundation of Mathematics.

Glenn said...

John,

One problem is that this view seems incapable of accounting for talk about the class of all real numbers, since human minds could never construct them all.

Human minds can't imagine all of them, at least not each and every last one, true. But that hardly rules out either our ability to grasp the concept of the class of all real numbers or our ability to make true statements about it.

Glenn said...

John,

Another problem is that it seems to entail that assertions about very large numbers (in particular, numbers that no one has thought about) are all untrue.

I have no idea how this conclusion is arrived at.

For, if no one has ever mentally constructed some very large number, then according to this view any proposition about that number will be vacuous.

I also have no idea how thisconclusion is arrived at. If n is allowed to be any positive integer, who would deny that one more than that positive integer, i.e., n+1, is greater than n? and why would claiming that n+1 is greater than n be "vacuous"?

A third problem with saying that mathematics has being in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.

How might a human grasp mathematical entities, work with, use and apply them if they haven't any being in his mind?

Glenn said...

s/b "..why would the statement 'n+1 > 1' be "vacuous"?

Daniel said...

@John West,

Admittedly this might display a pre-contempoary understanding of philosophy of Mathematics but I find it hard not to look at that discipline as having a strong parallel with Logic, particularly the Principles of Identity and Non-Contradiction, even if it’s nothing like the reduction to the former the Logicists wanted (this one of the reasons why I suspect the notion of a world were Mathematics differs is incoherent). So treating numbers as univocal seems odd to say the least.

It may well make sense to say that when the claim 'Socrates is Good' and 'God is Good' were are assuming the term 'Good' in an analogues sense but have trouble seeing how in the claim 'Socrates is One' and 'God is One' the term 'One' can have anything but a univocal meaning.

Alternatively in the propositions 'The Number of the Great Tragedians is Three' and 'The Number of the Persons is Three' are we using 'Three' in an analogues or univocal sense? I would be interested to hear what the official Thomist answer is here.

John West said...

Glenn,

You know, I read that link from Maurer just the other day. Thanks for the reply.

I'll start with your replies to my earlier arguments. Since I'm writing somewhat quickly, my replies may come off as abrupt—feel free to read them in a funny accent or something to blunt that unintended abruptness:

Another problem is that it seems to entail that assertions about very large numbers (in particular, numbers that no one has thought about) are all untrue. For, if no one has ever mentally constructed some very large number, then according to this view any proposition about that number will be vacuous (this was one argument).

Algebraic proofs are about all numbers. In mathematics, that's what it means to be a proof. Yet, if the relevant “very large numbers” exist only as mental objects (or entities) and no one has mentally constructed them, the relevant “very large numbers” don't exist. If they don't exist, there is nothing for algebraic proofs, which are intended to refer to all numbers, to refer to for the relevant “very large numbers”.

I wrote: A third problem with saying that mathematics has being in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.

You replied: How might a human grasp mathematical entities, work with, use and apply them if they haven't any being in his mind?

I should have written “A third problem with saying that mathematics has being only in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.” If we're discovering a mathematical entity that exists apart from the human mind, then its truth is objective and not contingent on psychological facts. If not, then it exists only if people decide to mentally construct it.

John West said...

Glenn,

Concerning the linked article:

There are important consequences of Aquinas's placing the notions of mathematics in the second order of quaestio disputata instead of the first. Unlike concepts on the first level, those on the second do not properly exist outside the mind. Their proper subject of existence is the mind itself. They are not signs of anything in the external world. Hence mathematical terms cannot properly be predicated of anything real: there is no referent in the external world for a mathematical line, circle, or number.[1] Finally, mathematical notions are not false; but neither are they said to be true, in that they conform to anything outside the mind. Aquinas does not suggest that they might be true in some other sense.

This claim seems to relegate mathematics to either nonsense—not even worthy of a truth value—or to breach the law of the excluded middle[2]. The author later concludes similarly:

Are all—or at least some—of the objects of mathematics real beings, or entia rationis, or do they lie somewhere between real beings and entia rationis? The present study leads to the conclusion that they are all constructs of the mind, but they have real remote foundation in the sensible world.

Assuming this statement does not deny the law of the excluded middle, this quote isn't saying that mathematical entities have external, non-mental reality, but rather that they're based on and abstracted from the physical world. This is basically a physicalist response, somewhat reminiscent of Mill's philosophy of mathematics. Moreover, these abstractions would have to stay fairly close to their physical exemplars to even have claims to having a foundation in physical reality (ie. an infinite line is quite different from a line segment; never mind yet other abstractions we can make about lines from line segments by copying symmetrical shapes and the like, which wouldn't even be similar after a point).

Say instead we assume that Maurer is just doing a somewhat poor job actually writing about Aristotelian realism. Then, since my quibble was about non-physically instantiated mathematics, that's another conversation and I have my answer as it concerns my original quibble.

Though, on Maurer's linked work, I'm not sure what role Scholastic realism plays. It seems to just collapse into old fashioned Aristotelian realism.


[1]I added the [1]. This footnote isn't in the original. But for instance, upon closer inspection of any physical shape or “circle” (say with a zoom function or a really powerful scope of some sort), find that it isn't actually a circle—that there are breaks in the circle, uneven parts, etc. Similarly, Euclidean lines, which are infinite in both directions, cannot be physically instantiated because physical reality is finite.
[2] I'm not making a cheap point here. There is actually a prominent—if on my view somewhat silly—school in philosophy of mathematics called intuitionism that talks about mathematics as mental constructs and fullstop denies the law of the excluded middle.

John West said...

Glenn,

Sorry, I forgot to italicize this quote in my first reply:

Another problem is that it seems to entail that assertions about very large numbers (in particular, numbers that no one has thought about) are all untrue. For, if no one has ever mentally constructed some very large number, then according to this view any proposition about that number will be vacuous (this was one argument).

That was my original statement, to which you replied.

John West said...

Daniel,

Alternatively in the propositions 'The Number of the Great Tragedians is Three' and 'The Number of the Persons is Three' are we using 'Three' in an analogues or univocal sense? I would be interested to hear what the official Thomist answer is here.

Incidentally, realist structuralists would say no. They're not different. It's merely one instantiation of three, which at bottom refers to positions in an abstract pattern.

I think James Franklin, the best and possibly only defender of Aristotelian realism in mathematics, also has structuralist sympathies. I haven't yet read his book, but I strongly recommend his work in philosophy of mathematics to every person not yet familiar with it interested in a fully fleshed, Aristotelian mathematical realism.

---

Also,

If we're discovering a mathematical entity that exists apart from the human mind, then its truth is objective and not contingent on psychological facts. If not, then it exists only if people decide to mentally construct it.

“[...] then the corresponding mathematical truth is objective and [...]”

Anonymous said...

Facts about God can only be known analogically. What about the fact that facts about God can only be known analogically?

Glenn said...

John,

I'm still not clear on why the existence of "Plato's world" is deemed to be necessary in order that, e.g., the mathematical statement 'n+1 > n' not be "vacuous". This seems to be what you're effectively saying, and, honestly, it makes little sense to me. It seems to me that that statement, and any other mathematical statement, can be, and indeed is, quite meaningful--whether "Plato's world" exists or does not exist.

Whether a mathematical notion enters the mind via a connection with Plato's world, via abstraction from sensible things by the intellect, or even by way of massaging, manipulating or extrapolating or inducing from prior abstractions (from sensible things by the intellect), that mathematical notion is in the mind. How it gets there does not alter the fact that it is there.

If Plato's world does exist, and a mathematical notion in the mind is not received from it but instead is abstracted from sensible things, would you say that that mathematical notion is for that reason "vacuous"?

John West said...

Glenn,

I wrote:

*Another problem is that it seems to entail that assertions about very large numbers (in particular, numbers that no one has thought about) are all untrue. For, if no one has ever mentally constructed some very large number, then according to this view any proposition about that number will be vacuous (this was one argument).

and:

**Algebraic proofs are about all numbers. In mathematics, that's what it means to be a proof. Yet, if the relevant “very large numbers” exist only as mental objects (or entities) and no one has mentally constructed them, the relevant “very large numbers” don't exist. If they don't exist, there is nothing for algebraic proofs, which are intended to refer to all numbers, to refer to for the relevant “very large numbers”.

You replied: I'm still not clear on why the existence of "Plato's world" is deemed to be necessary in order that, e.g., the mathematical statement 'n+1 > n' not be "vacuous". This seems to be what you're effectively saying, and, honestly, it makes little sense to me. It seems to me that that statement, and any other mathematical statement, can be, and indeed is, quite meaningful--whether "Plato's world" exists or does not exist.

Concerning the topic of your second paragraph, I wrote:

A third problem with saying that mathematics has being in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.

You replied: How might a human grasp mathematical entities, work with, use and apply them if they haven't any being in his mind?

I replied: I should have written “A third problem with saying that mathematics has being only in minds is that it makes mathematical facts contingent on psychological facts, which is absurd.” If we're discovering a mathematical entity that exists apart from the human mind, then its truth is objective and not contingent on psychological facts. If not, then it exists only if people decide to mentally construct it.

You further replied: Whether a mathematical notion enters the mind via a connection with Plato's world, via abstraction from sensible things by the intellect, or even by way of massaging, manipulating or extrapolating or inducing from prior abstractions (from sensible things by the intellect), that mathematical notion is in the mind. How it gets there does not alter the fact that it is there.

I honestly don't understand how I'm being unclear. I don't understand where communication is failing.

John West said...
This comment has been removed by the author.
John West said...

Glenn,

Concerning your third paragraph. If the terms in a sentence fail to correspond to anything in reality, then those terms are vacuous and statements using them aren't true. Consider the sentence:

All asdlkweiriowqaalskdf are green

Is that true? If not, why? Crudely, I'm saying it's because asdlkweiriowqaalskdf doesn't “pick out” anything in the world (construed as “all reality”). Similarly, propositions about fictional entities like (say) Tauntauns run into the same problem. So:

If Plato's world does exist, and a mathematical notion in the mind is not received from it but instead is abstracted from sensible things, would you say that that mathematical notion is for that reason "vacuous"?

No. I wouldn't. But in that case it wouldn't be abstracted from sensible things in the full, Aristotelian sense. I would be “discovering” mathematical structures instantiated in physical matter.

Incidentally, I wasn't necessarily making arguments for Plato's world. I think the Scholastic response (to which I'm partial) that the Forms are reflections of the Divine Essence, or Ideas in the Divine Mind is also fine[1].

[1]assuming we can actually talk about those.

Glenn said...

John,

Incidentally, I wasn't necessarily making arguments for Plato's world. I think the Scholastic response (to which I'm partial) that the Forms are reflections of the Divine Essence, or Ideas in the Divine Mind is also fine[1].

This seems like a good point on which to end the discussion. Thanks for making it interesting.

John West said...

Glenn,

Right. I don't think this is any problem for Scholasticism or classical theism (or divine simplicity). That's why I called it a quibble with Thomism, and its take on analogy/simplicity.

Thank you for the informative discussion.

John West said...

Daniel,


Admittedly this might display a pre-contempoary understanding of philosophy of Mathematics but I find it hard not to look at that discipline as having a strong parallel with Logic, particularly the Principles of Identity and Non-Contradiction, even if it’s nothing like the reduction to the former the Logicists wanted (this one of the reasons why I suspect the notion of a world were Mathematics differs is incoherent). So treating numbers as univocal seems odd to say the least.


I think a neo-logicist reduction to logical realism just kicks the problem upstairs.

John West said...

Another problem with saying mathematics has no extra-mental reality that I had forgotten, is that it seems then no one could ever really be wrong. Suppose you say 2+1=3, and I say 2+1=4. How would you know that I'm wrong, what standard would you measure it by? If mathematical psychologism is correct, then in John West's mind 2+1=4 could very well be right, even though in Glenn's (or whoever's) 2+1=3 is correct.

If one takes out their jelly beans and starts counting, then they effectively concede mathematics has at least some objective, extra-mental reality. In short, if mathematical psychologism is the case, then it seems mathematics could only ever be true by convention. Yet there are more peer corrections in academic papers in math than any other subject.

Of course Aquinas would say mathematics does have at least some foundation in extra-mental reality, so this is just another reason I find pre-Fregean mathematical psychologism so outrageous.

Glenn said...

John West,

If one takes out their jelly beans and starts counting, then they effectively concede mathematics has at least some objective, extra-mental reality.

Suppose I want to check whether "2+1=3". I take out my jelly beans, and start counting. After patting me on the back for my commendable exercise of self-restraint -- (I counted them, rather than ate them) -- you make two claims:

1. "In counting the jelly beans, you effectively conceded that mathematics has objective reality."

2. "Not only that, in counting the jelly beans you effectively conceded that the objective reality of mathematics is extra-mental."

I agree with your first claim, but disagree with your second claim.

When I disagree with your second claim, however, I am not disagreeing that the objective reality of mathematics is extra-mental, only that my counting of the jelly beans effectively conceded that the objective reality of mathematics is extra-mental. Had I been clear on how engaging in a physical act to check a mental understanding effectively concedes existence to something extra-mental, I might have agreed instead.

Another problem with saying mathematics has no extra-mental reality that I had forgotten, is that it seems then no one could ever really be wrong. Suppose you say 2+1=3, and I say 2+1=4. How would you know that I'm wrong, what standard would you measure it by?

The body of axioms, definitions, postulates, theorems, etc., of mathematics, out of which may be abstracted that from which it can be shown that 2+1=3 rather than 4 (in any base larger than base 3).

John West said...

Glenn,

A quick reply (busy week):

1. "In counting the jelly beans, you effectively conceded that mathematics has objective reality."

2. "Not only that, in counting the jelly beans you effectively conceded that the objective reality of mathematics is extra-mental."

I agree with your first claim, but disagree with your second claim.


I want to keep replies in their proper place. In the argument to which you reply, I was only making the first claim and not the second claim (even though the second is in quotes). That argument is against pre-Fregean mathematical psychologism, not a further claim. In fact, all four of my arguments have been against pre-Fregean mathematical psychologism.

On the condition that we're talking about mathematical psychologism (the idea that mathematical singular terms refer to mental objects existing in human minds) still, I'll respond to the rest:

The body of axioms, definitions, postulates, theorems, etc., of mathematics, out of which may be abstracted that from which it can be shown that 2+1=3 rather than 4 (in any base larger than base 3).

I can deconstruct this reply and further replies until you answer “due to convention”, if you want. In other words, I can reply to each of the listed in the same manner I replied to 2+1=3. The problem with a conventionalistic, psychologistic approach is that it fails to explain the applicability of mathematics to the extra-mental world.

only that my counting of the jelly beans effectively conceded that the objective reality of mathematics is extra-mental. Had I been clear on how engaging in a physical act to check a mental understanding effectively concedes existence to something extra-mental, I might have agreed instead.

Well, my point was that if mathematics has no extra-mental reality, then you shouldn't be able to check your mental understanding of mathematics using physical reality at all.

John West said...

only that my counting of the jelly beans effectively conceded that the objective reality of mathematics is extra-mental. Had I been clear on how engaging in a physical act to check a mental understanding effectively concedes existence to something extra-mental, I might have agreed instead.

If you mean to reply more like Maurer and not like the mathematical psychologist, that mathematics is merely a modeling language for the extra-mental world, then one problem comes from indispensability arguments:

People ought to have ontological commitment to all the entities indispensable to our best scientific theories—that is, our best theories about physical reality. Mathematical entities are indispensable our best scientific theories. Hence, people ought to have ontological commitment to mathematical entities. To quote Putnam, it is “intellectually dishonest” to have ontological commitment to some entities posited by our best scientific theories, but not the other entities of and required for those theories to work.[1]

And I refuse any reply like “there may be another modeling language no one has found yet.” It's the anti-realist's job to show we can dispense with reference to mathematical entities in our best scientific theories, similar to how it was Scott Bakker's job to show that eliminative materialists can dispense with intentionality in eliminativism.


[1] Obviously the argument is mainly aimed at scientific realists. But Michael Resnik points out that even if mathematical anti-realists are instrumentalists about scientific posits, like electrons, if mathematical entities are still indispensable to their methodology, then they still ought to have ontological commitment to mathematical entities. In showing they can dispense with mathematical entities in our best scientific theories, mathematical anti-realists cannot in any way use language referring to mathematical entities.
[2] Incidentally, if creatio ex nihilo can be proven philosophically, I think this can be used to argue these claims point to Scholastic Realism, the Divine Essence, and God.

John West said...

"In counting the jelly beans, you effectively conceded that mathematics has objective reality."

2. "Not only that, in counting the jelly beans you effectively conceded that the objective reality of mathematics is extra-mental."


Wait a second, I misread this quote. My comment was:

If one takes out their jelly beans and starts counting, then they effectively concede mathematics has at least some objective, extra-mental reality

That's what I was saying.

Glenn said...

John West,

1. >> The body of axioms, definitions, postulates, theorems, etc., of mathematics, out of which may be abstracted that from which it can be shown that 2+1=3 rather than 4 (in any base larger than base 3).

> I can deconstruct this reply and further replies until you answer “due to convention”, if you want.

After ruling out my being compelled against my will to give that answer, there is little chance you, or anyone else, will hear it from me.

But I'm puzzled that you have made your displeasure with anti-realism rather clear, yet seem willing to adopt (what on Balaguer’s account would be) the fictionalism version of anti-realism to wave away the response.

2. In An Aristotelian Realist Philosophy of Mathematics, to which you provided a link in a prior comment to Daniel, Franklin writes that, "Fregean Platonism about logic and linguistic items has...contributed to a distorted view of the indispensability argument, widely agreed to be the best argument for Platonism in mathematics."

The problem, as Franklin sees it, is that, while, "[i]t is obvious that mathematics (mathematical practice, mathematical statement of theories, mathematical instruction from theories) is indispensable to science, ...the indispensability argument arises from more specific claims about the indispensability of references to mathematical entities (such as numbers, sets and functions), concluding that such entities exist (in some Platonic sense)." [First emphasis is Franklin's; second emphasis mine.]

Franklin goes on to say that, and here I paraphrase, there is no reason to believe in a Platonist entity X simply because references to entity X cannot be eliminated from our language.

(If one considers that last too quickly, then Franklin almost -- almost -- sounds somewhat like Bakker talking about intentionality. Of course, there is a huge difference between Franklin's careful attention and Bakker's flippant sloppiness. (And, of course, Franklin is not saying that the non-eliminable nature of references to X isn't reason enough to believe that X-entities exist, only that it isn't reason enough to believe that said entities exist as Platonic entities.))

3. You haven’t necessarily made arguments for Plato’s world, and I certainly haven’t argued for anti-realism. The only thing which lured me into this discussion was the apparent stance that if one doesn’t subscribed to a full-blooded Platonism then one necessarily must be an anti-realist (and the worst kind of anti-realist at that).

The primary object of my curiosity in this discussion has been: why must that might be the case? That is, why must there be two extremes without any middle ground?

John West said...

Glenn,

After ruling out my being compelled against my will to give that answer, there is little chance you, or anyone else, will hear it from me.

But I'm puzzled that you have made your displeasure with anti-realism rather clear, yet seem willing to adopt (what on Balaguer’s account would be) the fictionalism version of anti-realism to wave away the response


I haven't adopted any fictionalism. Instead, I've been attacking and defending against two possible anti-realist positions, to be thorough.

I think maybe what's happened here is that you're really defending Aristotelian realism, which is mathematically realist, but that the link to Maurer confused me. I don't find Maurer clear at all. In fact, it seemed to me Maurer presents Aquinas like a Millian physicalist, not an Aristotelian realist.

I was just covering all my bases against anti-realism.

2. In An Aristotelian Realist Philosophy of Mathematics, to which you provided a link in a prior comment to Daniel, Franklin writes that, "Fregean Platonism about logic and linguistic items has...contributed to a distorted view of the indispensability argument, widely agreed to be the best argument for Platonism in mathematics." 

The problem, as Franklin sees it, is that, while, "[i]t is obvious that mathematics (mathematical practice, mathematical statement of theories, mathematical instruction from theories) is indispensable to science, ...the indispensability argument arises from more specific claims about the indispensability of references to mathematical entities (such as numbers, sets and functions), concluding that such entities exist (in some Platonic sense)." [First emphasis is Franklin's; second emphasis mine.]


Yeah. I think what James Franklin is getting at here is the Quinean tradition of using (Frege's) first-order predicate logic with identity as a canonical language. As Franklin points out in this paper, the problem with Quine's canonical language is that it begs the question against universals (Platonic, Aristotelian, or the Scholastic solution). It only lets you quantify over particulars, which is why Quine held to the existence of free floating, abstract, mathematical particulars.[1] But the indispensability argument is actually only reliant on three theses (fewer for Resnik's version): the indispensability thesis, confirmational holism, and the thesis that science is our best source of knowledge about the physical world[2]. None of those have anything to do with Quine's idiosyncratic canonical language methodology.


[1] This very mistake is why you sometimes get people like Balaguer complaining that his realist theory can't explain the applicability of mathematics to physical reality, when no classical Platonist would have ever had any problem explaining the applicability of mathematics to reality. Of course, Quine had other reasons for his canonical language, but I agree with Franklin on his point about the canonical language—that's why I've been careful to write “entities” and leave the form of the realism as a further debate.

It's also whence the whole “problem” of God and abstract objects comes, which is interesting since Quine himself thought there were no necessary beings at all — not even his mathematical particulars. Quine had his reasons, but I think the problem of God and abstract objects is a non-problem.
[2] Though this is actually stronger than needed. Even if you deny this thesis so stated, no one denies science is still at least very close to our best source of knowledge about physical reality and one of our best sources of knowledge about reality.

John West said...

3. You haven’t necessarily made arguments for Plato’s world, and I certainly haven’t argued for anti-realism. The only thing which lured me into this discussion was the apparent stance that if one doesn’t subscribed to a full-blooded Platonism then one necessarily must be an anti-realist (and the worst kind of anti-realist at that). 

The primary object of my curiosity in this discussion has been: why must that might be the case? That is, why must there be two extremes without any middle ground?


Much earlier, I wrote above: Say instead we assume that Maurer is just doing a somewhat poor job actually writing about Aristotelian realism. Then, since my quibble was about non-physically instantiated mathematics, that's another conversation and I have my answer as it concerns my original quibble.


My initial query was about uninstantiated mathematical entities and Scholastic Realism, the latter of which I have been led to understand accounts for uninstantiated mathematical universals (this is supposed to be what distinguishes it from old school Aristotelian realism). Since I do think we need to account for the truth of at least a significant amount of uninstantiated mathematics and consider any theory that doesn't do this incomplete, I'm really interested in deciphering all the fine points of Scholastic Realism (especially as it may pertain to dealing with problems in contemporary philosophy of mathematics). So one reason I may have seemed to not care about Aristotelian realism here is because my query was about Scholastic Realism and a detail of how it accounts for uninstantiated mathematics, which Aristotelian realism denies, in the first place.

Incidentally, I do have problems with old fashioned, non-Scholastic Aristotelian realism (ie. I'm unsatisfied with Franklin's reply to the perfectness objection, but still need to read his book), which I consider a far more serious contender than any anti-realism, and that's a debate I'm more than willing to have at any time. It just wasn't my concern this time.

John West said...

I should probably have written holism instead of confirmational holism, since I think I read the other day that there are different types of holists in philosophy of science.^

John West said...

Sorry, one other quick point. I wrote "non-physically instantiated mathematical entities" also to leave the door open for Platonism as well as Scholastic realism, to avoid picking a side, but clearly uninstantiated mathematical entities are also non-physically instantiated mathematical entities. It's just a more general way to write about it.

John West said...

On second thought, "non-physically instantiated" is unclear. I should have written "not-physically instantiated". I apologize for that lack of clarity.

John West said...

"[...] for that [sloppy, misleading mistake]"^

Scott said...

No sloppy misleading mistake, just an ambiguity in English hyphenation: non-physically instantiated is a natural antonym to physically instantiated, except for the unfortunate fact that in print it looks like it means instantiated non-physically.

It's too bad we don't have any standard way to group phrases together and stick a hyphenated prefix in front, as in non-(physically instantiated).

Scott said...

(I personally didn't even find it unclear; I knew exactly what you meant.)

John West said...

It may also be worth mentioning that when contemporary philosophers of mathematics talk about Platonism, they almost always mean Quine's "Platonism" and make no effort to distinguish it from classical Platonism. It's infuriating.

John West said...

Scott,

It's too bad we don't have any
standard way to group phrases together and stick a hyphenated prefix in front, as in non-(physically instantiated).


That would be nice.

Glenn said...

John West,

I haven't adopted any fictionalism.

Your question [1] and subsequent response [3] to my answer [2], which wasn't offered as a 'story', was eerily reminiscent of what Balaguer has to say about fictionalism and what a fictionalist might do (see pp 46-47 in his Philosophy of Mathematics (starting with, "One obvious question that arises for fictionalists is...").

Nonetheless, and obviously, there is a difference between being willing to do something and actually doing it, and I hadn't said that you have adopted fictionalism, or even that you are willing to adopt it -- only that you [had] seem[ed] willing to adopt it (and only for one particular, specified purpose).

[1] "Suppose you say 2+1=3, and I say 2+1=4. How would you know that I'm wrong, what standard would you measure it by?"

[2] "The body of axioms, definitions, postulates, theorems, etc., of mathematics, out of which may be abstracted that from which it can be shown that 2+1=3 rather than 4 (in any base larger than base 3)."

[3] "I can deconstruct this reply and further replies until you answer 'due to convention', if you want."


- - - - -

Btw, my use of 'extra-mental', as in the jelly bean caper above, hasn't been in the sense of existing outside or external to the mind, period (which I gather is the way you've been using the term), but in the sense of existing in a non-instantiated state in some 'realm' higher and above the mental (which 'realm' is higher and above the physical 'realm'). I think my understanding and usage of the term in a sense differing from the sense in which you understand and use it has led to us partly talking past one another. I had had inkling early on that this might be the case, but I didn't seek to clarify it. That is my fault; sorry.

John West said...

Glenn,

In context, the quote for your [1] is Another problem with saying mathematics has no extra-mental reality that I had forgotten, is that it seems then no one could ever really be wrong. Suppose you say 2+1=3, and I say 2+1=4. How would you know that I'm wrong, what standard would you measure it by? If mathematical psychologism is correct, then in John West's mind 2+1=4 could very well be right, even though in Glenn's (or whoever's) 2+1=3 is correct.

If one takes out their jelly beans and starts counting, then they effectively concede mathematics has at least some objective, extra-mental reality. In short, if mathematical psychologism is the case, then it seems mathematics could only ever be true by convention.


My point was that for adherents of pre-Fregean mathematical psychologism statements like 1+2=3 can only be said to be any more true than 1+2=4 by convention. I can apply the same argument to “The body of axioms, definitions, postulates, theorems, etc “ I don't think this was unclear from what I wrote.

Btw, my use of 'extra-mental', as in the jelly bean caper above, hasn't been in the sense of existing outside or external to the mind, period (which I gather is the way you've been using the term), but in the sense of existing in a non-instantiated state in some 'realm' higher and above the mental (which 'realm' is higher and above the physical 'realm'). I think my understanding and usage of the term in a sense differing from the sense in which you understand and use it has led to us partly talking past one another. I had had inkling early on that this might be the case, but I didn't seek to clarify it. That is my fault; sorry.

Ahh, so you were thinking of extra-mental as not-known-by-the-mind's-eye in the Platonic sense maybe? Fair enough. I completely missed that possibility. Thank you for clarifying.

[Edit: Too many late nights turning coffee into theorems recently]

John West said...

not known-by-the-mind's-eye-in-the-Platonic-sense^

Glenn said...

[Edit: Too many late nights turning coffee into theorems recently]

I used to do that with algorithms. (And having to attend to such basic necessities as food and sleep was a constant source of resentment. Having to eat was a major nuisance because the stomach needs blood to digest the food, and that blood has to come from somewhere, and the brain was always being conscripted to supply some of it, which meant less oxygen to the brain, which in turn meant a decline in the clarity of thoughts; and sleep was feared -- feared, I say -- coz who knew if all that in the mind would there or capable of being reconstructed upon awakening.)

;)

John West said...

I used to do that with algorithms. (And having to attend to such basic necessities as food and sleep was a constant source of resentment. Having to eat was a major nuisance because the stomach needs blood to digest the food, and that blood has to come from somewhere, and the brain was always being conscripted to supply some of it, which meant less oxygen to the brain, which in turn meant a decline in the clarity of thoughts; and sleep was feared -- feared, I say -- coz who knew if all that in the mind would there or capable of being reconstructed upon awakening.)

Haha. So you understand exactly then!

[Posted in the right thread this time, too]

Glenn said...

Haha. So you understand exactly then!

Yup.

Scott said...

@Glenn:

"[W]ho knew if all that in the mind would [be] there or capable of being reconstructed upon awakening."

I know that feeling all too well, with respect to mathematics, computer programming, and law. I have to say, though, that one of my two best grades in law school was in a class on Constitutional law, in which we downloaded the final exam (at our convenience during exam week) from the professor's website and then had twenty-four hours to return our answers via email. I waited until a suitable bedtime during exam week, downloaded it, read it over, and then went to sleep. When I woke up in the morning I'd worked everything out in my sleep and I just had to write it out.

Glenn said...

Scott,

That's very interesting. I've had the odd experience of working out solutions to bugs while sleeping. (It's odd, but most welcomed; saves a lot of time.) Were you aware while sleeping that you were working things out ('sleep mentation')? Or basically just taking advantage of knowing via prior experience that, somehow, things sometimes get worked out while asleep?

John West said...

Scott,

When I was younger, one of my AMC camp instructors told us to do math until right before sleep. He claimed the brain continues going over material during the night. Maybe he wasn't just trying to motivate us. Maybe there was something to it.

Scott said...

@Glenn:

"Were you aware while sleeping that you were working things out ('sleep mentation')?"

I was, and I'd had previous experience with that sort of thing that induced me to try it on my final exam as soon as the opportunity became available. Of course I'm not saying it's the best night's sleep I've ever had, but ordinarily in life one has to take only so many Constitutional law exams.

@John West:

"Maybe there was something to it."

I'm sure there is. That's not the only time this sort of approach has worked for me.

John West said...

Glenn,

An aside. I just read the linked Balaguer pages. The difference between the relevant mathematical psychologists and Balaguer is that the psychologists wanted to claim that mathematical singular terms referring to mental objects existing in human minds made mathematical statements true. Balaguer just bites the bullet and says they're untrue (well, he says false, but he waves his hands about it), except in the fictional story of [...].

Balaguer's fictionalism doesn't explain the applicability of mathematics to non-fictional reality. But Balaguer claims this is no problem, because neither does his full-blooded (Quine-like) Platonism (FBP), which he considers the only tenable version of mathematical realism. I don't consider it the only tenable version of mathematical realism. But anyway, Balaguer is in a different situation from the old mathematical psychologists I was arguing against.

Incidentally, since this is a links thread: Portraying Analogy. Do either of you know if Ed has written about Portraying Analogy anywhere?

John West said...

Here it is free: James Ross, Portraying Analogy

John West said...

An aside. I just read the linked Balaguer pages. The difference between the relevant mathematical psychologists and Balaguer is that the psychologists wanted to claim that mathematical singular terms referring to mental objects existing in human minds made mathematical statements true.

Say, in my earlier example, that in John West math not only is 2+1=4 true, but 2+1≠3 is also true.

Assume the psychologism is true. Some mathematical statements are true in a realist sense (their claim). Some identical mathematical statements are false in the same sense (what I've shown). Identical mathematical statements cannot be both true and false. Therefore, the psychologism is false. Hopefully that clears it up.

Glenn said...

John West,

Put that way, yes, it's clear. (And I ought to have inferred as much earlier, but, for some inexplicable reason, I didn't.)

- - - - -

Re Ross' Portraying Analogy, Dr. Feser has linked to it before (here), but I'm not aware of any lengthy comments he may have made on it.

Glenn said...

Oh, look at that.

In his Suggestions on How to Study (February 6, 2015), William Vallicella quotes from Alphonse Gratry's "The Sources of Intellectual Light" (1862):

"Set yourself questions in the evening; very often you will find them resolved when you awaken in the morning."

Scott said...

@Glenn:

Well, great golly goodness gosh. Couldn't really be much more on point, could it?

John West said...

Glenn,

"Set yourself questions in the evening; very often you will find them resolved when you awaken in the morning."

Sure didn't hurt, whoever thought of it first.

Not too sure about waking at 2 AM, though. Bill Vallicella's a machine.