Philosophy of Mathematics

user21820

05:52

@user170039: I will continue our discussion about realizations here, because as I said earlier I would cease discussion with Heinrich if he erroneously insists that ZFC is inconsistent. Plenty of cranks have done this before, but no legitimate logician has yet found an actual inconsistency. Some cranks have done worse, claiming that PA is inconsistent...

I know of only one legitimate logician, Edward Nelson, who sincerely believed that PA was inconsistent, and published a 'proof', but later retracted his claim when his error was pointed out.

The relevant comments are:

"Russell's barber" is a concept that has no instantiation. Namely, there is no such situation (village plus barber). It is not in any way a paradox except if one makes inconsistent (worse than unsound) assumptions such as naive set theory. Note that it is perfectly possible to have a set theory with a universal set (NFU) or a type theory with a universal type (the first-order theory of Turing machines where each TM is also the type of all TMs that it accepts). The latter is concrete and shows clearly that a universal type is not inherently problematic. — user21820 20 hours ago

By instantiation I mean an object of reality that is defined by the definition. Since words are real objects they can be instantiations too. Your notion of instantiation is too restrictive. Would a barber with a missing leg be an instantiation of "male barber"? Would a barber with a missing penis be? Concerning set theory: It is a self contradictory nonsense, not worth mentioning. — Heinrich 18 hours ago

The text "Heinrich" is not the same entity as the Heinrich I'm now talking to. "Heinrich" is something that I can put on a piece of paper. Heinrich is someone I can't physically touch, much less put anywhere. The two do not have the same properties. Similarly the mere phrase "Russell's barber" is just a phrase and is not a barber of any sort. Concerning set theory: it's not self-contradictory even if it is nonsense, but you're not qualified to judge that. — user21820 18 hours ago

@user21820: What do you mean by "realization" here? Can you give an example? — user 170039 15 hours ago

@user170039: Suppose we believe that in reality we can construct and store binary strings in some specific physical medium that represent natural numbers, and suppose we further believe that we can perform arithmetic operations on any given binary strings. Then we can call the whole collection of these strings as a realization of "model of PA". — user21820 15 hours ago

So far as I understand Heinrich's definition of instantiation is related to making sense of the definition (and hence is related to the cognitive abilities of the human(?) mind to understand the meaning of sentences and/or phrases). And so he/she says that, "Russell's Barber has the instantiation "Russell's Barber". Because we all know what is meant by this definition,.." In other words, for him/her a concept has instantiation if the lingual expression of the concept makes sense to us (feel free to object Heinrich if you don't think this is what you intended). — user 170039 2 hours ago

But in your case (so far as I have understood) a concept has an instantiation if it is possible to have a real situation where there is a concrete object which (seemingly) corresponds to the concept under consideration. Am I correct @user21820? — user 170039 2 hours ago

The rest of Heinrich's comments are irrelevant nonsense.

I shall address your last comments and inquiries.

user21820

06:13

It is clear that Heinrich (if he is consistent) considers the phrase "Russell's barber" to be an actual barber, because that is one of the properties ascribed to the concept of "Russell's barber". Nobody here disagrees that the concept of "Russell's barber" is an actual well-defined concept. However, Heinrich conflates syntax and semantics in the typical illogical way, and does not appear to realize it.

Also, Heinrich clearly implies that he thinks "Russell's barber" actually defines an object of reality (see the quoted comment above). Any logical person can see that it is false. Remember that this concept of "Russell's barber" has nothing to do with set theory, but laymen (likely including Heinrich) think that it does.

For another example, "A natural number whose square is 2" is a meaningful concept, represented by the 1-parameter arithmetical sentence P where P(n) ≡ ( n*n = 2 ), but has no instantiation/realization, and this fact is represented by the theorem ¬∃n ( P(n) ).

If you doubt the theorem (and hence must doubt PA), it is also represented by the empirical fact that every physical numeral (say binary strings as described in my other comment above) when squared (via the corresponding physical realization of multiplication) does not result in the physical instantiation of 2, in every working computer on this planet. This is a highly predictive statement and hence cannot be dismissed as chance or triviality.

So yes your last comment is basically correct. There is no mathematical way to refer to any physical interpretation, since all mathematics are symbols and have no way to 'reach out from the page into the real world', so to speak.

But if you curtail our natural language to a formal language that still has an inbuilt notion of "real-world entity" and can reason about them in the appropriate ways, then you can even formalize all my comments above.

user21820

06:47

@MikhailKatz I have 4 points for you. I do not think you can logically disagree with them. (0) You have not provided a meaningful syntactic representation of 1/3 in the hyperreals. (1) Nobody can legitimately disagree that hyperreals can be constructed via the ultraproduct of the reals R within ZFC, which is the mainstream foundations for mathematics.

(2) Philosophically nobody has provided non-circular ontological arguments justifying ZFC (especially with replacement and choice). No logician, whether on Math SE or on Math Overflow or whom I have met, have done anything close to it. I accept various things such as consistency of ZF implying consistency of ZFC, but consistency is quite irrelevant to soundness besides being necessary. Unless you're happy with just Π1-soundness.

(3) The construction of the hyperreals is via the ultraproduct of the reals R. If you can construct the hyperreals, then you also can construct R and prove the usual second-order real axioms for R. It would be self-contradictory to say that the properties of R (including 0.999... = 1 suitably interpreted) are not intuitive and then claim that the hyperreals are intuitive. After all, we define an infinitesimal in the hyperreals as a nonzero sequence of reals that converges to zero...

@user170039: Please see my last point (3). Informally, it implies that one cannot claim that hyperreals are philosophically meaningful but reals are not. And that is despite closing both my eyes to the use of ultrafilters on N... If you don't know what those are, then you don't know what the hyperreals are... It is consistent with ZF that hyperreals do not exist. What makes you think an ultrafilter on N philosophically exists?

This point is nearly orthogonal to our previous discussion, except that it can also be phrased as follows:

(2b*) You do not accept as philosophically justified any foundational system that is needed to construct the reals (I don't care what aspect about it you do not accept) and hence cannot accept as philosophically justified any foundational system that is needed to construct the hyperreals.

I am of course ignoring the trivializing way out, which is to have a foundational system that just blindly assumes the existence of hyperreals. In that case, I have further objections to the misuse of "0.999..." as suggestive of hyperreals:

(@MikhailKatz: These points are open-ended unlike my previous 4, so there's nothing to agree/disagree with, but if they cannot be addressed adequately, I'll stick to my assessment of hyperreals as separate from the intuitive misconception that 0.999... < 1.)

user21820

07:21

(4) Let R* be the hyperreals and ε = 1 − 0.999... You claim that ε is nonzero in a suitable interpretation of 0.999... Ignoring the fact that you cannot represent 1/3 meaningfully in similar decimal form, I now present you another fact that you can't represent ε/2, not to say sqrt(ε). Wait, what does the latter even mean in the hyperreals. Can your students figure that out? Are you sure hyperreals are so intuitive now?

In contrast, asymptotic expansion can happily deal with sqrt(x) for any asymptotic expression x that is non-negative. No trouble at all. x^(1+x) for positive x? No problem.

Mikhail Katz

11:38

@user21820 I find this format not particularly convenient. How about if we copy this to either a question or answer at MSE and take it from there?

user21820

@MikhailKatz As I said before, I'm personally not interested in the hyperreals until someone can justify the chosen foundational system in which he/she constructs them. So I can't be bothered to post about the hyperreals. My point (0) is there only because you in particular claimed in your very own answer "Instead of taking its limit (which is by definition real-valued), one can take what Terry Tao refers to as its ultralimit, to obtain a number than falls infinitesimally short of 1.".

My point shows that it is philosophically less satisfying (more weird) to adopt such a possible interpretation of 0.999...

A Q&A on Math SE would be even less suitable, since mathematically I already said I know precisely how to construct the hyperreals, so we're talking philosophy not mathematics.

In other words, my 4 points above are in response to you, not an inquiry that I am personally invested in.

@MikhailKatz: I'll repeat my stand; provide me philosophical justification of ZFC or any other foundational system that can construct the hyperreals, and then we can talk about their meaningfulness or usefulness.

Mikhail Katz

12:03

@user21820 I wasn't suggesting creating a separate thread but rather copying this to an existing question or answer, for example mine at user 170039's question.

I just find this format with its length limitations very awkward.

user21820

@MikhailKatz That's really unsuitable; you know that comments on Math SE are not for prolonged discussions, and they too have length restrictions

@MikhailKatz You can get around the restriction. Just type what you want into a text editor and then copy it wholesale into the message box.

As long as there is at least one new-line, it allows you to send it and it will appear in a box with "(see full text)".

Mikhail Katz

12:18

@user21820 I was thinking of modifying my answer there, not working with comments.

user21820

12:57

@MikhailKatz You're of course free to modify your answer as you wish, but please don't drag me into any discussion by mentioning me or my comments in this chat in your answer. Since the issue here is philosophy, I'm not interested in any further discussion about hyperreals until someone justifies ZFC philosophically.

user131753

@user21820 What do you mean by justification of $\sf{ZFC}$ "philosophically"?

user21820

@user170039: I've given you the links to my post about replacement before, with all the relevant comments from some logic experts on Math SE. I presume you haven't read them, so you can start there.

user131753

@user21820: I tried to go through it. But it is way above my knowledge. Can you phrase it in non-technical terms?

user21820

@user170039 I can try to, but what roughly do you not get? Just ask in the Phil of Math room.

user131753

@user21820: I mean an example of "justification of $X$ philosophically" by 'it' in my previous comment.

user21820

13:10

Um, this notion is vague because I'm very lenient and permit any kind of justification as long as the assumptions it relies on seem more obviously meaningful to me than the conclusion. For example of what I do not permit: justifying induction on N via well-ordering on N, or vice versa.

You can see from the post I linked plus the comments that it is circular justification that I do not accept.

Even if you can't understand the technical details, and even if you don't buy my claims, at least know that the well-known logician Boolos claimed the same thing.

user131753

@user21820: I asked this question because it is not at all clear to me what you mean by philosophical justification of $\sf{ZFC}$ and hence what kind of justification you are asking from @MikhailKatz.

user21820

@user170039 I understand this, but all set theorists and logicians know what I'm talking about.

(I presumed he is one.)

I can give you some positive examples of what I'm looking for.

user131753

@user21820 Do you mean ontological justification of $\sf{ZFC}$ as philosophical justification of $\sf{ZFC}$? The epistemological justification? What is the guarantee that such a justification indeed is possible?

user131753

@user21820 I don't understand what you wanted to say here. Can you clarify?

user131753

@user21820 If they really know then they should be able to give us at least an approximate idea of what they count as Philosophical Justification and what they don't.

user21820

13:25

@user170039 In my linked post I explicitly cited Boolos, who said the same thing as I did in my answer. Originally I just posted my answer based on direct logical analysis of the claimed ontological justification for replacement as some people have given, and explained why it was circular. However, it seemed clear that a couple of readers did not have the logical facility to understand the circularity, and I happened to later find about the part I quoted from Boolos.

I don't know how much clearer it can be if you don't understand ZFC and transfinite induction...

Boolos says:

> This bounding or cofinality principle is an attractive further thought about the interrelation of sets and stages, but it does seem to us to be a further thought, and not one that can be said to have been meant in the rough description of the iterative conception. [...] Thus the axioms of replacement do not seem to us to follow from the iterative conception.

This implies that the iterative conception cannot serve as an ontological justification for ZFC, since it does not imply the replacement schema.

@user170039 The problem is that we can see and explain clearly what is circular, but we cannot necessarily tell you what a reasonable ontological justification might be. Perhaps there is one that we have yet to think of. But up till now no logician has seemed to have done it.

@user170039 There is no guarantee, and in my opinion ZFC cannot be justified to have real-world meaning.

I've stated this many times from the beginning. That's why I keep bringing it up, because people keep using things that ZFC can prove, but that I don't believe is meaningful.

If we want to talk philosophically, then you can't use anything that I don't find meaningful unless you give me reason that it is meaningful!

By "you" I don't mean "you personally", by the way. I mean "anyone".

@user170039: Also, I want to explain why I linked that post. I explicitly state what philosophical assumptions that correspond to what rules or axioms as we climb the hierarchy of weak foundational systems. So you can see for yourself how one may justify those systems ontologically.

By making those assumptions explicitly clear, one can see whether or not one accepts those justifications.

user131753

9

A: Are sets and symbols the building blocks of mathematics?

The things you actually write on the paper or some other medium are not definable as any kind of mathematical objects. Mathematical structures can at most be used to model (or approximate) the real world structures. For example we might say that we can have strings of symbols of arbitrary length,...

user21820

@user170039: Perhaps you could try reading the following post first, whose first half was written for a student audience (to justify induction):

4

A: Why are induction proofs so challenging for students?

Logic foundation In my opinion, the only way for anyone to really understand induction is to really understand the logical structure behind it. So a prerequisite is a complete grasp of working in first-order logic, for which I recommend both boolean algebra and natural deduction (Fitch-style) in...

user131753

1

A: Why is replacement true in the intuitive hierarchy of sets?

Suppose that as each stage $S$ is completed, we take each $y$ in $x$ which is formed at $S$ and complete the stage $S_y$. When we reach the stage at which $x$ is formed, we will have formed each $y$ in $x$ and hence completed each stage $S_y$ in $\mathbf S$. This is actually a circular justi...

user131753

13:48

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user131753

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user131753

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user131753

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user21820

@user170039: Concerning the post about induction, it is interesting that it can serve both as a philosophical justification of induction and as a mathematical justification of the soundness of adding induction to any ω-consistent formal system, if suitably translated. A system S is said to be ω-consistent iff ( for every 1-natural-parameter sentence P, if S proves each of the sentences P(0), P(1), P(1+1), P(1+1+1), ..., then S does not disprove (the suitable translation of) ∀n∈N ( P(n) ).

Sorry, I mean consistency of adding induction to any ω-consistent formal system.

user21820

14:29

To justify soundness of adding induction to S, you need S to be sound. This may not make sense if S talks about things that have nothing to do with reality, but we can still have an intermediate notion; ω-models. The terms "0", "1", "1+1", "1+1+1", ... are called standard numerals. M is said to be an ω-model of S iff every natural in M interprets some standard numeral, in which case we say that M has no non-standard naturals. Adding induction to S preserves its ω-models.

I've to take back temporarily what I said about adding induction to ω-consistent formal systems preserving consistency. The naive argument I thought would work doesn't seem to work, so I'll have to think about it more.

Transcript for

♦ $Mathematics$ Philosophy of Mathematics

Transcript for

♦ Philosophy of Mathematics

♦ $Mathematics$ Philosophy of Mathematics