In his first
Meditation, Descartes famously tries to push doubt as far as he can,
in the hope of finding something that

*cannot*be doubted and will thus provide a suitable foundation for the reconstruction of human knowledge. Given the possibility that he is dreaming or that an evil spirit might be causing him to hallucinate, he judges that whatever the senses tell him might in principle be false. In particular, the entire material world, including even his own body and brain, might be illusory. Hence claims about the material world, and empirical claims in general, cannot in Descartes’ view be among the foundations of knowledge.
Surprisingly
for a rationalist, Descartes also suggests that even claims about basic
arithmetic cannot be among the foundations.
For he proposes that it is possible that God might make him go wrong
when considering even something as elementary as the claim that 2 + 3 = 5 or
the claim that a square has four sides. By
the end of the first Meditation, he alters the scenario so that it is the evil
spirit or Cartesian demon rather than God who is doing the deceiving, and later
in the Meditations he argues that given God’s perfect goodness, he cannot be
leading us astray in any way. The key
point for present purposes, though, is that Descartes does suggest, at least
initially, that it is as coherent to doubt basic arithmetical and geometrical claims
as it is to doubt that one is awake or that matter exists. Is this true?

I think
not. The way Descartes’ skeptical
scenarios work is by proposing coherent alternatives – or purportedly coherent
ones, anyway – to the way things appear to common sense. For example, given the experiences you are
having right now, your common sense assumption would be that you are looking at
a computer screen reading a blog post.
However, there are, Descartes says, clearly possible alternative
scenarios in which you are not really looking at a computer screen and reading
a blog post at all. You could instead be
in bed asleep and having a vivid dream about reading a blog post on a
computer. Or you could be a disembodied
spirit who is being caused by a Cartesian demon to hallucinate that you have a
body that is sitting in front of a computer reading a blog post.

Put to one
side for present purposes the question whether these particular scenarios
really are, at the end of the day, coherent.
(Some philosophers have
argued that they are not.) They are at least prima facie plausible
insofar as we are familiar enough with dreams and hallucinations. It is possible to have a dream in which one is
convinced one is awake and looking at a computer screen. It is possible to hallucinate. Hence it certainly seems like we have cases
where things appear to be some way X but are really some other way Y. To be sure, one can (and should) question
whether it is coherent to suppose that one is

*always*dreaming or hallucinating, or whether a sensory experience of*precisely the kind I am having right now*could really be a dream or hallucination. But dreams and hallucinations are familiar enough that these particular skeptical arguments at least get off the ground, even if we can ultimately shoot them back down.
By contrast,
it is not clear how skepticism about basic arithmetic can even get off the
ground. For what we need is a coherent
scenario in which it seems that (say) 2 + 3 = 5 but in reality the arithmetical
facts are very different. For example,
we need a coherent scenario in which 2 + 3 = 14 but God or the demon is making
it seem otherwise. And the problem is
that there is no such coherent scenario.
We simply cannot coherently describe a case in which 2 and 3 really add
up to 14, the way we can (arguably) coherently describe a case in which you are
not really reading a blog post on a computer right now. For 2 and 3 adding up to 14 is a logical impossibility,
whereas your not really reading a blog post on a computer right now is

*not*a logical impossibility. The proposition that*2 + 3 = 14*entails contradictions, whereas the proposition that*I am not really reading a blog post on a computer right now*does not.
You might
respond: “But maybe God or the demon is only making it

*seem*to you to be a logical impossibility.” But that won’t work, for the same reason the original scenario won’t work. For now we need to be able to describe a scenario in which the proposition that*it is logically possible that 2 + 3 = 14*is itself logically possible (and either God or the demon is only making it seem otherwise). And the problem is that*that*proposition is no more logically possible than the first one is.
There is,
then, a crucial disanalogy between the examples involving arithmetic and the examples
involving physical objects. We can,
independently of what we know about dreams and hallucinations, make sense of a
scenario in which a certain physical object is not present. (There are, after all, a great many places
devoid of computer screens and blog posts – my back yard, the surface of the moon,
the bottom of the Mariana Trench, etc.) Hence
we can go on to contrast a dream or hallucination in which the object does at
least

*seem*to be present with the fact that it is not. But we*cannot*independently make sense of a scenario in which (say) 2 + 3 = 14. Hence we have nothing to contrast with a scenario in which God or the demon makes it seem as if 2 and 3 add up to something other than 14. We can’t really get the skeptical scenario going, the way we can with skeptical arguments involving dreams and hallucinations about the physical world.
So, it seems
that, even

*if*Descartes were correct to regard skepticism about the senses and the material world as coherent, he should not have regarded skepticism about basic arithmetic and the like as coherent. That is significant for the rest of his project in the Meditations. It has often been pointed out that, given the latter sort of skepticism, Descartes arguably shoots himself in the foot, making it impossible for him to get beyond the Cogito and maybe even impossible to get as far*as*the Cogito. For if I could be wrong even about something as seemingly self-evident as 2 + 3 = 5, why couldn’t I be wrong about something like*Cogito, ergo sum*(“I think, therefore I am”)? Or, even if I can’t be wrong about that, I still need to carry out some fairly complex reasoning to get from knowledge of my own existence to knowledge of God’s existence (as Descartes does later in the Meditations, before going on to appeal to God’s goodness as guarantor of the reliability of his rational faculties). And how can I be sure that I haven’t gone wrong somewhere in that reasoning, if I can be wrong about something as basic as 2 + 3 = 5?
It is a good
thing for Descartes’ overall project, then, that the doubts raised in the first
Meditation vis-à-vis basic arithmetic and the like are misplaced. (That’s not to say the project isn’t wrongheaded
in

*other*ways – it is – but at least this bit of it can be patched up.)
Further
reading:

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ReplyDeleteBut in that case "0" and "1" are no longer the same objects we are accustomed to. Sure, we may be using the same names and the same symbols to represent them, but in reality they're not the same concept anymore.

DeleteThough, in the end, I think that the point you raise is the reason why some mathematicians give in to nominalism, thus rejecting the much more common Platonism. I remember when I first started college level mathematics several years ago I was enchanted by the axiomatic construction of the natural and the real numbers.

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DeleteThe operation being conducted in Galois theory on a field of characteristic 2 is not arithmetical addition. Your counterexample depends on an equivocal use of the + sign and is therefore out of court.

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Delete

DeleteHow do you know they're not the same 0, 1 as in (Z, +)?Fields of characteristic 2 violate the Peano axioms (and any of the other formalizations of common sense arithmetics, which are equivalent to the Peano axiomatization). Meaning that the objects you just constructed don't have the same properties as the natural numbers, and hence cannot be the same as them.

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DeleteHow exactly does (R,+) violate the Peano axioms? Sure, they only formalize arithmetics over the natural numbers, not over the real numbers, and when talking about the group (R,+) we're not considering the operation of multiplication, but that doesn't mean (N,+) considered as a subgroup of (R,+) violates the relevant subset of the Piano axioms.

DeleteIt just means that for present purposes we're not considering all the usual structures one usually imposes on the real numbers as a way to model common sense arithmetics.

Cogniblog, one can talk of (R, +) as an extension of (N,+), or that (N,+) is a substructure of (R,+) under the standard injection of N into R. Under such an injection N is seen as a subset of R and the addition operation is preserved, so we can consider them the same operation.

DeleteThat the operation is effectively the same between two structures need not imply that the same first order statements will hold, because the truth of the statements in structures are relative to their interpretation within the underlying set.

Correction: I wanted to say that (N,+) is a subsemigroup of (R,+); (Z,+) would be a subgroup, but strictly speaking the Peano axioms only apply to N (including 0). Anyway, the point of the argument stands.

DeletePhilosopher Hillary Putnam made an argument against the claim that the world world might be an illusion.

DeleteWe can call the trees that we see in the real world trees, and those we see in the illusion world **trees** Now if you think that you are seeing trees when you are actually seeing **trees**then is this really a mistaken judgment? You might think that **trees** aren't real, but you don't seem to have any reason to think that trees aren't real. Since you have only ever seen **trees** and never actually seen any trees, you are not really casting doubt on the belief in trees. You might suspect that **trees** are't real, but what reason could you have for thinking that trees aren't real?

@Cogniblog

ReplyDeleteI might be wrong, but I think the point Ed tried to convey is that we can not doubt something that entails logical contradiction. So when you have example like 2+2=14 that is meant just as example of something that is logically contradictory; it does not mean to introduce complex math but just meant that if you have two object and two objects you will get 4 objects not 14 objects (and what you mean with say number 14 is just abstracting what is common to 14 cats, 14 dogs, ...).

"Your lecturer says that in fields of characteristic 2, 1 and 1 do indeed make zero. "

Obviously it is not same binary operation of addition as in the case of integer numbers Z, because it maps same pair of numbers to different number. So lecturer did not show that 1 plus 1 is not 2 but only that 1+1=0 (+ is here equivocal; + used is different + then before, we can call it "plus bar" if that makes you happy) in field of characteristic 2.

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DeleteWhat we know about 1 and 2 and addition, as objects, in this context necessarily entails that 1+1=2. Logically, it is necessary.

Delete"They just believe that addition behaves differently in different contexts."

So you are not using addition in the same context, then? Doesn't that just sidestep the point, rather than refute it?

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DeleteBut it is logically necessary that the concepts of 1 and the concepts of 2 and the concepts of addition as understood by the audience Feser was addressing in this post necessarily lead to the logical conclusion thst 1 and 1 make 2.

DeleteYou might as well say that in certain concepts 1 is actually a cipher that means "A", + means the word "and", = means "also", so it really means "a and a is also aa", e.g., 1+1=11.

This is technically all true but very obviously misses the point.

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DeleteYou're just changing the subject. We knew exactly what Dr. Feser meant when he used the symbols he did. Given our agreement on what those symbols represent 1 plus 1 equals 2.

DeleteIf you think it means something else you're just talking about something else.

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DeleteNow you're being obnoxious. Grammatically the "just" was not an attempt to simplify your arguments but to point out that you're not addressing the substance of Dr. Feser's point. You're still not. If we all know what the numbers 1 and 2 are and what the symbols plus and equal mean, then 1+1=2

DeleteYes, if you mean something different by the numbers and symbols then 1+1=2 could be wrong. But it nobody here means something different.

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Delete@Cogniblog "if 2 + 3 = 5 is a tautology, why does it require computation?"

DeleteWhy would it matter if it takes computation to verify a statement?

Also, what do you mean by "context"? From what you've said, a context is a set of definitions that you use to interpret a mathematical statement. But if that's the case, then interpreting "1+1=2" and "1+1=0" with different contexts would be equivocating. Fancy equivocating, but still equivocating.

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DeleteFor clarification, in what sense are you using the term tautology? The mathematical definition is that a tautology is a statement that is true for all structures. For example, "for all x, x =x". But having to compute something doesn't make it not a tautology. For example, in a proofs class you might be asked to show that p -> (q -> r) <-> (p & q ) -> r. This is, at least, by mathematical definition, a tautology, but it isn't necessarily obvious.

DeleteOr do you mean something different by "tautology"? Are you working more along the lines of a tautology as a definitional statement, hence immediately true, but containing no information? (ex" the overused "a bachelor is an unmarried man")?

I do think you are on to something, reading your other posts here (especially your example with the alien race who thinks in terms of fields of characteristic 2). If Descartes is questioning the intuitional mathematical framework we in practice operate under (i.e., the specific mathematical structure of arithmetic with N, or Euclidean geometry with R), then yeah, that's fair game.

However, I think part of the problem is that mathematics itself implicitly only makes conditional claims. The statement is not "1+1 = 2". But rather, "if we are in N and addition is defined as such-and-such, and if we assume peano's axioms, then 2+3=5." In practice, we tend to assume the antecedent without even mentioning it, because the antecedent is our intuitional framework. However, the strength of that statement is not the strength of said framework, but rather the strength of logic itself. It's been a while since I read Descartes, but I doubt he was in the business of questioning logic, since his work was a sort of exercise in logic to begin with.

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DeleteFar, far back in this thread a question was addressed to me. So:

DeleteIf 1 and 1 and 2 mean what everyone here agrees they mean, then 1+1=5 is not a tautology, it is wrong.

If you are saying they can mean something different, fine. But we're not discussing something different, we're discussing the uses of those numbers Dr. Feser was clearly referring to.

Also, it is rich of you to call us out for reading you uncharitably when you called out a response of mine for using the word "just" in it, as if I was attempting to restate your arguments incorrectly when I wasn't even attempting to do that.

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DeleteThat's great that your postdoc said that. It also wasn't what I was doing.

DeleteBut anyway, moving on. It isn't a tautology, unless you're using a different definition of the word than I know.

I'm saying that for the phrase "1+1=2" to be true, we all need to understand those symbols to mean certain things. We all do. Therefore when 1 is added to 1 you get 2.

That is what Dr. Feser was referring to, not possible alternate definitions of those words.

Unless I'm misunderstanding Dr. Feser's argument. I don't think so, though.

I actually think Tony's objection below is more interesting - that in a dream we may be completely convinced we have a proper understanding of such and such mathematical equation/principle/what have you without realizing that our understanding CAN'T be right even in principle.

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ReplyDeleteBy contrast, it is not clear how skepticism about basic arithmetic can even get off the ground. For what we need is a coherent scenario in which it seems that (say) 2 + 3 = 5 but in reality the arithmetical facts are very different.I don't seem to understand the argument here. When I was studying Euclid or later math, a couple of times I dreamed that I had a proof of a theorem, but when I woke up I had no proof. What seemed, in the dream, was a proof, was only the illusion of a proof. Now, it is my impression that what happens in the dream is not that one actually thinks A, then B, then C, then D, where (because of the dream) each seems to follow from the previous: you don't actually think "D follows logically from C". Rather, I think that what happens is that in the dream you have the internal feeling that occurs when you do in fact experience thinking B follows from A, and C from B, and D from C: you have the feeling in the dream of the conclusiveness or validity of several steps. You don't (at least in my experience) actually reason from A through B and C to D, and falsely think D follows from C, you just think

jumbled stepsand then "there, the theorem is proven". The illusion isn't in thinking precisely A, B, C, and then D, it is in thinking "the conclusion is proven" when all that happens is a mish-mash of half-formed thoughts that aren't even about D to begin with.So, the "coherent scenario" isn't a

plausible argumentthat people imagine might in some weird way substantiates that 2 + 3 = 14, but a plausible scenario that people areunder the illusionthat they have experienced a proof of the thesis 2 + 3 = 5, when they all they have experienced is thefeelingof certainty that (ought to) come from validation through a proof. Since that feeling can be mimicked in a dream, the plausible scenario is that they have the feeling of certainty from a dream.Now, I still don't necessarily think this argument works. For one thing, if my description is accurate to what people experience, they DON'T experience clearly perceived theses with clearly apprehended concrete intermediate steps in such dreams, they experience jumbles and vague, amorphous ideas that are not specific enough to be capable of proof anyway. So when we ARE being precise and specific, that is not like a dream. Now, Descartes can claim that this is just a "super-duper dream that is

specialbecause it is better than the ordinary class of dream", but effectively that's just special pleading. Or, he could argue that the demon can give us clear and concrete ideas about which he induces that feeling of certainty (and the feeling "it was validated in a proof", and off the cuff I don't know what response would be given.I think one can only doubt that 2+3=5 if one doesn't understand the concepts of 2 and 3 and addition. If 2+3=5 is just a strings of characters you memorized in kindergarten to get a piece of candy, I can understand why the truth of that mathematical statement would seem arbitrary. In reality, the symbol "2" is just a convenient marker we use to refer to the concept of 2. I'm not sure you can prove "concept of 2" "concept of addition" "concept of 3" "concept of equality" "concept of 5". It's just something you "see" with your intellect once you've gotten a certain understanding in mathematics.

ReplyDeleteHey Ed,

ReplyDeleteI think the relevant question is, "Is it possible for one to affirm a logically incoherent proposition with an ultimate intensity of conviction?" and, based upon experience, the answer seems to me to be, "Yes." People are honestly mistaken--though (at least for a time) adamant concerning their rectitude--quite frequently when working through problems in arithmetic or logic . . .

If one doubts the logical necessity of mathematics, one undermines the logical laws she used to come to the conclusion that 2+3 can = not 5. Atleast, as far as I can tell.

ReplyDeleteThis comment has been removed by the author.

DeleteThe scenario doesn't have to involve denying the logical necessity of mathematics. Mathematics is logically necessary, but perhaps your understanding or grasp of basic mathematic truths could be wrong (Feser argues it isn't)

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DeleteSomething could be tautology to advanced intellects while being not to common humans.

DeleteAs CS Lewis remarks somewhere, God is Love is tautology to angels but not to men.

DeleteSo it would be a tautology like saying the Queen of England is English.That's not actually a tautology.

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DeleteThere are several different things that are meant by 'tautology', depending on what you are doing; on the content-free version, such as some Wittgensteinians and others accept, nearly everything else you've said about tautologies in this comments thread is wrong -- what has no content can be neither true nor false, so in accounts that treat 'content-free' as a property of tautologies, tautologies are not logically necessary truths, nor are any of the examples you've given of tautologies actually tautologies. On a structure-invariance account (such as that given by Mary Angelica elsewhere in the thread), some of your examples are similarly perplexing. Most of the things you've said look like they require an analytic account of tautology, in which case it's false that no conclusions can be drawn from them, and Gyan's point applies -- one of the handful of things we learned from logical positivism in the twentieth century, if we didn't know it before, is that something can be a tautology, in the analytic sense, in one context and not another, because it depends on the available definitions and axioms. If you are using the term in a different way from any of these three, though, it's not surprising that people might be struggling to know what you are saying, because most people would be assuming some version or other of one of the three.

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DeleteWittgenstein is not the only one who holds such a view; the reason he was mentioned is that the overwhelming majority of people who think that tautologies are content-free are Wittgensteinian about tautologies. There are no other widely accepted accounts of tautologies in which they are content-free. The private language argument is, of course, entirely irrelevant to the question, even if you had correctly described it; the account is not based on the private language argument.

DeleteIn any case, you continue your streak of apparently switching from one account to another while pretending to be giving one account. If there is no content, there is nothing that can be true or false to begin with, on any major theory of truth, whether deflationary theory, correspondence theory, coherence theory, or pragmatic theory; all of them directly require content for truth assignment. Thus on a content-free account, tautologies are not true or false; they are rules, or empty forms to be filled with content, and thus not even the right kind of thing to be true or false.

A=A is only capable of being logically true on a structure-invariance account (in which case it identifies content that is invariant regardless of structure) or on an analytic account (in which, usually, it would have the definitional account relevant to equality).

As for your assertion that "nearly everything else you've said about tautologies in this comments thread is wrong,"You need to learn how to read more careful and not misquote the people to whom you are responding. I did not assert this; I asserted that it followed from standard content-free theories of tautology that almost everything you've said about tautologies is wrong, and I explicitly gave the reason: on all widely accepted content-free accounts, tautologies are not the right sort of thing to be true or false, whereas you have been assuming they are true. I did not assert that you were wrong at all; I asserted that if one assumes a particular account of tautology you were. The distinction between asserting a claim and asserting that a claim follows from something else is an elementary logical distinction, and you would sound more like you know what you are talking about it if you took more care to observe it in interpreting others.

Thus your response does not address the issue: there are standard families of accounts of tautologies on the table that are widely accepted -- content-free, structure-invariance, analytic -- and none of them are consistent with all of your claims about tautologies. So there are only two possibilities here: either your view of tautologies is incoherent, or you have some other, nonstandard, not widely accepted view of tautologies, which you are assuming and which you keep not explaining to anyone else, who are all most likely to assume one of the major theories. Thus, assuming you are not an idiot, it is not surprising that people don't understand what you are saying, because you keep leaving out essential information that they would need to know -- namely, what you mean by your nonstandard use of words.

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DeleteI do believe that Brandon is a philosophy professor. I’ve been following him for years. Probably best to assume he needs you to spoon feed him though.

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ReplyDeleteWell, the questioj seems to be why can't squares have five sides and why can't 2 +3 equal 6 or why simply cannot coherently describe a case in which 2 and 3 really add up to 14. I agree that we can't, but why is that?

DeleteIOW what is the status of the laws of logic. "How" do they exist?

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DeleteI wonder how many arguments in my life I can settle by repeating emphatically that my whole point is demonstrated by the single word “transitivity” in italics.

Delete

DeleteOne can doubt 2+3=5 not because it contradicts common sense, but because it is incompatible with the axioms of geometry.I studied math, including geometry, a fair bit, and I have NEVER heard that "2+3=5" is incompatible with the axioms of geometry. I don't even have a guess as to what sub-branch of geometry one might look for such a thesis. I have heard of plenty of axioms of geometry, and narry a one mentions "2+3" to begin with, much less asserts 2+3 not= 5.

Pretty sure that "2+3=5" is compatible with the axioms of geometry. I think the correct assertion would have been:

"Rand=true" is incompatible with the axioms of logic, geometry, geology, physiology, numerology, and astrology. Not to mention car mechanics.

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DeleteIt's important to note that 1 + 1 = 2 is not true because of any psychological certainty or because it is "obvious". .9999 (repeating) is equal to one. Not because it is obvious or because we feel certain of it, but because it is the subject of a demonstration.

ReplyDeleteLikewise, it's not particularly important whether we can conjure up a psychological experience of doubt. What matters is whether we can determine the conditions under which a mathematical statement is true or false. Mathematics has the advantage that all its terms are defined systematically in relation to each other, and so the conditions for the truth or falsity of a claim are usually determined based on grasping definitions or relations without requiring further verification.

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DeleteYou'll have to explain yourself more. If it helps, I'm speaking about an arithmetic proposition.

Delete

DeleteAnd transitivity is not a demonstration?Transitivity is a word, denoting a property applicable to some operators. It is not applicable to all operators, nor is it applicable to bare entities like points, lines, or 3. Nor is it a demonstration of anything. If you would like to demonstrate something, you have to lay out starting theses (i.e. premises) and show conclusions follow from them. Nothing follows from "transitivity" because "transitivity" is not an assertion, it is a property.

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DeleteYou'll want to spell out an argument here. I suspect you don't have one.

DeleteThe "=" sign refers to equality. Of course there are different types of equivalence: the equivalence of expressions differs from the equivalence between functions, for example. As a programmer, I'm pretty familiar with the contextual complications that arise from equality not just in theory but in practice.

Transitivity is a property related to equality, but it's not the same thing. This is pretty obvious. Relations other than equality can be transitive (from inequalities (>) to set inclusion).

You're not only confused about Feser's point here, you're confused about math.

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DeletePhilip:

DeleteYou can't affirm Pythagorean theorem (a^2 + b^2 = c^2) and not accept the basic arithmetic laws governing summation. The Pythagorean theorem *assumes* those laws, as do the proofs for the theorem. None of this makes sense.

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DeleteWell, looking back on it I should have realized earlier this was not a serious conversation.

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Delete"You might respond: “But maybe God or the demon is only making it seem to you to be a logical impossibility.” But that won’t work, for the same reason the original scenario won’t work. For now we need to be able to describe a scenario in which the proposition that it is logically possible that 2 + 3 = 14 is itself logically possible (and either God or the demon is only making it seem otherwise). And the problem is that that proposition is no more logically possible than the first one is. "

ReplyDeleteBut if we are being forced to consider the possibility that our judgement of what is logically possible in the first place may be distorted, then i think we have no other option but to admit, that it might be the case, since we can not rule that out by pointing out that certain scenarios appear to be logically impossible.

Dr. Feser,

ReplyDeleteYou might like to see James Beebe's work on "a priori skepticism" in connection with this issue.

As soon as I read this blog post of Prof. Feser's, I knew immediately that the comment box would be cluttered with a batch of mathematical blowhardery that completely misses the point.

ReplyDeleteAnd, sure enough, here it all is on display.

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DeleteWe can apply all this to the writings of Dick Dawkins. In his "God Delusion" rantings, he said that the argument from religious experience is faulty because these experiences can be easily be dismissed as hallucinations. So Dick Dawk thinks that if an experience could possibly be a hallucination, then we should not trust it. But if we took that approach seriously, then we would be in the Cartesian situation where we cannot believe in the reality of ANY of our experiences, not even our ordinary sensory experiences. I'm sure Dick Dawk doesn't realize this, but I hope that one day, somebody will point it out to him.

ReplyDeleteWell, think about this: you can certainly doubt that 75832+95841=171673, until you check my calculations and notice that it is indeed the case. So, someone much stupider than you could certainly doubt even that 2+3=5.

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DeleteI don’t mean to insult him by claiming that he can’t quickly do 75832+95841=171673 in his head. I thought the calculation was big enough that most people can’t.

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DeleteWell, I fully expect I’m missing something here which makes me turn out to be wrong, but I don’t know what it could be. Thanks for the praise, I suppose.

DeleteMy guess is that here, you're not so much doubting that 75832+95841 = 171673 as you are that when you calculate 75832+95841, you will obtain the answer 171673. If you were to fully understand the operation—which is difficult for large numbers, but trivial for small ones—THAT would be something you could not coherently doubt. Similarly, I can imagine myself discovering a proof in a textbook that triangles can have more than three angles, but I am not thereby imagining that triangles in that scenario have more than three angles (something impossible to imagine).

DeleteI believe that if I have made a mistake, and I hope I have, then it surely is an equivocation about the “doubt”. I cannot quite put my finger on it, however.

DeleteSuppose I have a 75832mm long stick and a 95841mm long stick. I ask a stickmaker to make a stick that is as long as the sum of their lengths, and he gives me a 171673mm long stick. He certified it as being this length with marks along its body for every mm, and even gave me a note on the receipt stating that “75832+95841=171673”.

However, the stick seems to me rather too long, or maybe too short. It does not immediately seem to me to be the length I asked for, so I suppose that it is longer, or shorter, and the stickmaker made a mistake. It seems immediately coherent to me that this stick is not as long as I should have expected, but rather a different length. As it turns out, the stickmaker is a master at his trade, and it is not actually coherent, but it takes me a while to see this. I only notice after checking the numbers again, or maybe putting the sticks side by side.

Now let's suppose instead that I think that I am always hallucinating, or constantly dreaming, or I suspect that an evil demon has bewitched me. This seems to me immediately coherent, as Feser said. However, upon further reflection, suppose I eventually come quite conveniently to the exact same conclusion that Bouwsma did in the paper Feser linked, that is, that it is not actually coherent at all.

Now, in both those instances I was confused for a moment and thought that something incoherent was coherent, but then I thought for a bit and it turned out I was wrong. What is the difference between the instances, exactly? I cannot see it. If I understood the statements completely at first glance, I should never have been confused, and this is true for both cases, but that is not what happened. What is it that gives my confusion about the physical world a privilege over my confusion about maths?

To anyone coming later, please read further comments below, where Feser seems to ground the privilege on “simplicity of cognition”, thereby restricting it to the most basic arithmetics. The privilege is thus fully grounded and explained, though it does not exist in my exchange with the stickmaker, and our poor proverbial “someone much stupider” (later phrased as a “preschool child”) is still doubting our titular sum. Hopefully a tutor can help him.

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ReplyDeleteI don't mean the statement so much as I mean how it would be verified. In analogy, we can imagine a proof that there are not infinitely many twin primes, even if we later discover that there are (in which case it would be necessary).

DeleteMaybe a better way of putting it: if I were to doubt 2+2=4, I would be assigning the name "2+2" to a mathematical object that was not in fact 2+2, and hence obtaining a statement linguistically (and perhaps even cognitively) identical to 2+2=5, but not conceptually the same (which is what we really care about). If I somehow am convinced that "I" is just a word referring to a person with a set of properties that I happen to possess, I can use the word correctly and be right in 99.9% of cases, but I could still doubt "I exist" (i.e. imagine that the person I call "I" lost one of those properties and ceased to be). In doing so, however, I would not thereby be doubting that I exist, since the statement was only identical in a confused, purely linguistic sense.

DeleteIf inference is some mental process whereby I combine the concepts "2", "3", "plus" and "equals" to generate "5", why could the demon not intervene in that process just before I conclude "5" and befuddle me in such a way that I conclude "12." And then befuddle me again every time I attempt to bring together these notions to check my work?

ReplyDeleteI see that the demon would have to keep befuddling me every time this particular inference is implicated in some other inference, and so probably reduce me to state of permanent arithmetical befuddlement.

My intuition is that this wouldn't work without ultimately destroying my rational faculties, but I don't have a robust way of articulating this. After all, some people are, generally speaking, pretty bad at drawing logical inferences.

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ReplyDeleteI strongly advise you to read your quote as well as the "I answer that" part of that article you've taken it out from again

DeleteForgive me if this has been covered, but, has anybody asked Ed if he has time and/or interest in debating Jay Dyer on the argument from kinesis? Jay makes some good videos but seems overconfident in his dismissal of the argument... anyhow, hope everyone's well.

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DeleteLOL ... depending which video you catch, sure... the video containing the issue I mentioned is titled, "Simulation Theory & The Pagan Gods Refuted" ... it starts out with some heavy metal impersonation (I'm guessing this is what you were referring to?) but from roughly 40 minutes to the second hour is the "meat" I refer to.

DeleteI wonder why that “Cogniblog” guy has deleted all his comments.

ReplyDeleteGot the impression that everyone was thinking "look at that loony punk posting his stupid theories on numbers." Several people called me an idiot. I'm not gonna expend energy dealing with that.

DeleteI want to go on the record as not thinking you are an idiot. I do think you were missing the point, but you were doing so honestly and there's as always the chance I'm the wrong one. I'd have liked you to leave your comments up.

DeleteThe analog to the dreaming/hallucination argument for arithmetic: I sometimes make arithmetical mistakes. I might add 134757 to 383731 and get 517488, and so honestly believe that 134757+383731=517488 (even though the correct answer is 518488). Even though the belief that 134757+383731=517488 is inconsistent with the axioms of arithmetic, it is possible for me to believe it, because the inconsistency isn't immediately obvious.

ReplyDeleteCould I do the same thing for a really basic sum, like 3+4=7? Or 2+2=4? It doesn't seem very psychologically plausible to me. But, I've seen young children (preschool age) make such mistakes with these very basic sums. It is psychologically plausible that a preschooler might honestly believe that 3+4=6. So, if a preschooler can make such a mistake, could not an adult?

Maybe, due to some neurological disease, or drugs, or whatever, my psychology might become more infantile, such that mistakenly thinking that 3+4=6 or 2+2=5 suddenly becomes psychologically plausible once again? An intellectual reversion to early childhood?

But, if such a reversion is possible, how do I know I am not having one right now? Maybe my belief that 2+2=4 is actually the product of such a reversion?

Objection: if my mind had regressed in its arithmetical abilities to that of early childhood, I'd expect my verbal and philosophical abilities to also regress, and so I wouldn't be writing this blog comment right now.

Reply: is it possible that the regression might be selective, and my arithmetical abilities would regress to those of a preschool child, but my other intellectual abilities would remain at an adult level.

Objection: such a selective regression seems implausible. Do we have any evidence of it ever happening?

Hesitant reply: No such evidence, although brain injury sometimes results in weird cases of selective ability (certain skills being highly damaged while others are completely intact), so maybe a case of such selective regression might one day be produced.

I don't think the mistake you are discussing is equivalent to what Feser is denying. Yours is about unknowingly missing steps in the logical process, for whatever reason, such as not adding the 1. However, it assumes that if you follow it correctly, you would get the right answer. Feser is discussing the idea that even if you follow the process correctly, you could still come to the wrong answer, to which he denies.

DeleteI think you are also thinking this is about some general rules being misapplied. For instance, when people see, say, 102-33, many people can reflexively jump to thinking that the number will be somewhere in the 70-79 range, just because they will leap to the common rule that 10 and 3 relate to the number 7 in subtraction in someway. Thats not math, or even logic. That is just intuition. The preschooler might think that 2+3=6 precisely because they intuitively accepting it because they haven't quite grasped numbers and addition properly and are relying on memory, not because they have followed the process to that conclusion.

My “someone much stupider” argument from above might be much better phrased as your “preschool child” argument, but it’s still the same argument. It is nice to see that someone else saw merit in it.

DeleteNow, unfortunately for Billy, I don’t think Feser rested his argument on something as evident and uncontroversial as the fact that you can’t come to a wrong result without having made a mistake. No, because that is true for the external physical world as well: if I exist and you come to a wrong conclusion about this, then surely you have made a mistake somewhere.

Instead, Feser gives doubt about the physical world what I referred to in my previous comment as a “privilege”: he states that

regardlessof whether it is even coherent to think of oneself as being always dreaming, or constantly hallucinating, or deceived about the physical world by a devil, and therefore regardless of whether it is necessary to “skip a logical step” to think this (to paraphrase Billy), it is nevertheless more reasonable to doubt the physical world than mathematics.And this is because those scenarios are “at least prima facie plausible insofar as we are familiar enough with dreams and hallucinations”.

Don’t mathematical falsehoods seem plausible to us “prima facie” while we are making a mistake? This is before we catch the incoherence, that is, the fact that it entails a logical contradiction to think that 134757+383731=517488, and that it is impossible to construct a coherent scenario where this is true. This is the exact same thing that happens to incoherent beliefs about the physical world not existing, assuming they truly are incoherent. Thus, the privilege given by Feser to doubt about the physical world seems to be unjustified.

He clearly believes that it is possible to coherently describe a scenario where the external world does not exist, or that we cannot sense it if it does. Now, if he can defend this, and I believe he is capable of doing so, then all is well: external world skepticism is clearly superior to mathematical skepticism, since the former is coherent and the latter isn’t.

But if he cannot, then all he has to stand on is this “prima facie plausibility” privilege of doubt about the physical world over doubt about our knowledge of maths, regardless of coherence, and this does not seem to me defensible – or, I should say, I cannot see how it could be defended.

To anyone coming later, please read further comments below, where Feser seems to ground the privilege on “simplicity of cognition”, thereby restricting it to the most basic arithmetics. The privilege is thus fully grounded and explained, though it does not exist for the large sum at hand, and our poor proverbial preschool child (or “much stupider” man, as I had it) may still doubt our titular sum, which I believe is merely a minor exception that can not be further generalized to ground the skepticism of the able-minded.

DeleteLike Professor Pruss, I continue to doubt Professor Feser's argument on this.

DeleteLet me use as an example the tribe of primitives (in the Amazon jungle?) who do not have words or names for most numbers. They have one, two, and many. And that's it. Now, it is not fair to urge that THEY can doubt "2+3=5", because the doubt Feser is talking about is the doubt that holds when a person

understands the propositionbut can be indeterminate about whether it is a true or a false proposition. Tribesman Bill who doesn't understand the proposition doesn't have the first element of the required sense of doubt.No, my counterproposal is more involved. Let's take Bill and start TEACHING him higher numbers, with the aim of eventually being able to do math. First we must start teaching him numbers above 2, and we will have to take it slowly. Perhaps it takes some time before he gets "3" down, and then more time to get "4", and still more time to get "5" and "6". Let us suppose that he has been exposed to "7", but has not yet absorbed the concept, but that he has "6". Now we start to explain addition to him formally, not just the implicit addition in the counting upwards from 2. We do 1+2, 1+3, and 1+4. He has to work at it to get the

ideaof "plus", but work at it he does, and he gets it at least as a basic principle, although this is all so new he remains very much in dismay at all these new thoughts. So you have taught him his "1 plus" table up to 1+5, and now introduce the following questions:A: What is 2+2?

B: What is 2+3?

C: What is 3+2?

He has never tried any of the "2 plus" table identities, so these are brand new questions. At the moment he views the QUESTION, he know

what the question means, but he does not know the answer. He must think about it, and he might get it wrong. In fact, he knows HOW to get the answer toD: "What is 1+5"

from what he was taught in doing 1+2, 1+3, and 1+4, (rooted in counting) so he can go about (finding?, constructing?) the right answer for 1+5. But we have not yet given him any method for converting the problem "what is 2+..." into an equivalent problem of "what is 1+...", and that's the only sort of math he knows yet. So not only does he not KNOW the answer to "what is 2+3", he doesn't have the tools for coming up with the answer reliably. Yet. If you then asked "is 2+3=5", he would quite reasonably say "I don't know".

All this is just making clear and concrete what Dr. Pruss said above. Basic math or not, the statement is NOT "self-evident" and thus it is open to the possibility of doubt.

My own preferred "we cannot doubt" proposition, after we have gone beyond "I am", is this: "The whole is not less than the part". This proposition still requires a pre-existing set of understood concepts, including the right "less than" concept. But once they are understood, no person can be in doubt as to the truth of the proposition. (And it is much simpler than the statement of non-contradiction.) So, it's not like I dispute that there ARE such things that you cannot doubt - and that such can be used in Dr. Feser's point about "the demon making you think..." I just don't think "2+3=5" is a good candidate. I think Dr. Feser needs a proposition which is technically self-evident. (Note that moderns tend to dispute that there is such a thing as a self-evident truth other than a tautology, but Aristotle and St. Thomas clearly say there is such a thing.)

Hey Ed,

ReplyDeleteOff topic but have you seen Netflix's "Dark"? Interesting time paradox series.

Ed:

ReplyDeleteI think your argument proves too much. It is plainly possible to be reasonably skeptical of mathematical claims that are in fact necessarily true. If I were to multiply together two eight digit numbers, my confidence that the product is what I obtained should be low: it is not at all unlikely that I made a mistake. This is true even if in fact I happen to be right and there is no logically possible scenario where the answer is other than it is.

I think what this means is that Cartesian doubt shouldn't be taken to be grounded in the possibility of *this particular belief* being false. What exactly it's to be grounded in is hard to say, but roughly I'm thinking something like this: Could I have *this sort* of evidence while yet coming to a false conclusion?

In regard to the product of the two eight digit numbers, if I am in fact right about the product, I couldn't have *this* belief, or *this* evidence, and be wrong. But I could have *this sort* of evidence and be wrong.

Let's now think about *how* in fact we know that 2+3=5. Here are four different stories.

First, when we are small children, our evidence for it is something like this: If I count to two, and then count off three more, I get to five (using fingers helps with the process). But obviously I could miscount in this process.

Second, if you're a visual reasoner like me, you have this mental image of five objects, looking like it's divided into a group of two and a group of three. But I could imagine having this sort of evidence and yet being wrong. Maybe I have a mental image of a group of objects, but when I zoom in on a subgroup of it, the image shifts in a way that I don't notice. I have weirdly shifting mental images when I am tired, after all. So this sort of evidence could lead to falsehood.

Third, I also have it memorized. So I just rattle it off. Of course, we make mistakes in that sort of thing.

Fourth, we could go about this more rigorously. By definition, 2=1+1, 3=(1+1)+1 and 5=(((1+1)+1)+1)+1. The fact that 5=2+3 then follows from two (I think) applications of the distributive law. But this sort of evidence could lead to falsehood: it's easy to get mixed up in all the parentheses and all the +1s. And the reasoning behind the distributive law itself is complicated.

Rather than putting it in terms of psychological certainty, we should be thinking in terms of the conditions for certain judgment. If those conditions are met (as they often are in mathematics), then psychological doubt is not relevant. The example I gave above about .999 (repeating) being equal to 1 is one case where people often doubt the result even after the demonstration has been proffered.

DeleteThere's a deeper point about certainty, doubt, and reality here, though it's somewhat outside the scope of the original post. Many Thomists (maybe most of those on my reading list) fall into a form of naive realism where they put certainty about the reality of the world on the level of experience, either by some quality imminent in the experience or by some ineluctable psychological feeling of certainty possessed by the experiencing subject. Just as the feeling of doubt does not rule out the certainty attending an act of judgment in the case of mathematics, so it is with any judgment about the real world. Certainty is not attained by experience; reality as such is not an object of the senses. The answers to these questions are attained on the level of understanding and judgment, which may inquire into and appeal to experience, but operate on an entirely different level. This is the advance critical realism makes over naive realism.

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Delete@PhilipRand both statements are wrong. It's 0.999

Delete...= 1. The ellipses are significant. The reason why is because it 0.999...is definedas the infinite series0.999... = Σ_{k = 1}^∞ 9/10^k = 9/10 Σ_{k = 0}^∞ 9/10^k = 9/10[1/(1-1/10)] = 9/10 × 1/(9/10) = 1

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DeleteBut c isn't equal to b...

DeleteC is not equal to B, unless you have infinitely many zeroes, just like you needed infinitely many nines to make A equal B. And, well, yes, then everything becomes coherent: 0.000…00001 = 0.000…00002 = 0.000…0000823728957 = 0. And this is because having something “after” infinitely many zeroes is nonsense.

DeleteHi Alex,

DeleteNote that I repeatedly qualified my remarks by speaking, specifically, of "

basicarithmetic." So it wasn't the necessity of arithmetical claims alone that was doing the work, but necessity together with the simplicity of the cognition involved in elementary truths like 2 + 3 = 5, 1 + 1 = 2, etc. I agree that the point wouldn't apply to cases such as multiplying eight digit numbers.This comment has been removed by the author.

Delete@Phillip Rand

Delete@Thiago V. S. Coelho

Phillip, if by "1.0000000001" you actually meant to write "1.000000000...1", with the '...' standing for an infinite series of zeros, well then Thiago response is sort of right. However, I use the qualification "sort of", because in fact, there is no such number in the Real numbers (R). In R, there can be no limit number following the "..." representation.

With that said, you could identify the following well-ordered set:

2,3,4...1 to correspond to the infinite ordinal number w+1.

Likewise,

3,4,5...1,2 = w+2

But of course, infinite ordinal numbers are not members of the set of real numbers (R).

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DeleteWell, Feser’s last comment appears to me to grant the “someone much stupider” argument, a.k.a. the “preschool child” argument, since it consists in pointing out cases where there is no such “simplicity of cognition”. So, if I am stupid enough, I can certainly doubt that 2+3=5, though not otherwise.

DeleteNow, for the proverbial moron, there might be something even simpler which even he cannot doubt, such as that 5=5. If someone were stupid enough to doubt all of mathematical logic in this way, then we would probably not be speaking anymore of a mind intelligent enough to even doubt anything. So I suppose the “stupid preschool child” argument (as we might call it?) only presents a very limited set of exceptions, and not something that can be generalized.

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DeletePhilip, the stick example involves an error of perception, which is an error of the senses, not in the property of transitivity. How would Thomism be defeated by this?

DeleteDescartes seems to be dealing with a different kind of error.

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DeleteI don't understand principle 2. What do "commensurate" and "measurable" mean in this context?

DeleteEd:

DeleteThe simplicity here comes in degrees. There are lots of logical steps involved in proving 2+3=5. It would be odd if scepticism about things that involve, say, five steps were possible but scepticism about things that involve only one four steps were impossible.

The only reasonable place I could see placing a cut-off is where there is only one logical step. 5=5 may be like that (immediate application of the rule of identity introduction, and if you don't know how to use that rule, you don't know what "=" means). But 2+3=5 is definitely not a one-step inference (unless we're just relying on a memorized result).

DeleteThen by using the law of transitivity you come to the conclusion that bar A and bar C are the same length.

Sorites Paradox. And the answer depends on hysteresis.

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DeleteI think Feser is not claiming that skepticism about basic mathematics is impossible, but merely that it turns out to be incoherent. Skepticism about the external world, regardless of whether it is similarly incoherent, is awarded a privilege over skepticism about basic mathematics (it can “get off the ground”) because it is more “

Deleteprima facieplausible”, due to the “simplicity of cognition” involved in basic arithmetic. Failing to enjoy this privilege is said to mean skepticism about basic maths cannot “get off the ground”, but I do not take that to mean it is impossible. You can simply lie down on the floor and be a skeptic from there, until you get up and notice that you were lying down on a bed of incoherence.@PhilipRand you are talking about hysteresis. You are a Time Lord, so concepts like hysteresis come naturally to you.

DeleteThe answer is not that transitivity fails. The answer is that whether B = C depends on the history of the system. Even though you can't discern any difference between B= 1.0000000 cm and C= 1.0000001 cm, B is only equal to C when you're not moving from 1.0000000 cm to 1.0000001 cm in time.

Let me give you a Time Lord metaphor to help you understand. An air conditioner is set to turn on at 22 degrees centigrade. The ambient temperature is 22.00001 degrees centigrade. There are two cases.

The first case is that the air conditioner is turned out for the first time. It sees that it is 22.00001 degrees centigrade and cools. Here 22.00001 != 22.00000.

The second case is that the air conditioner already cooled down the room but the temperature rose to 22.00001 anyway. Here the air conditioner knows the history of the room, and says that 22.00000 = 22.00001

untilthe temperature rises to 25 degrees.In the Holy Trinity 3 = 1 metaphysically. Could God not create a reality where things are so different that (approaching things from an Aristotelian perspective) what we witness in that reality would in fact require a different mathematics?

ReplyDeleteI think you are mistaken about the Trinity. Now, God could also create a universe where reading is impossible, but that would not allow you to doubt you have just read this. So, your question is irrelevant to the point at hand.

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Delete> Really... but if God created a world where reading is impossible we would not have the Word.

DeleteThe Word of God is properly the Second Person of the Trinity, but I'm guessing you mean the written Bible. Luckily, transmission of the Bible does not require writing nor reading, though they are helpful tools. In many cultures, oral tradition is the ways stories are passed along from one person to another.