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Suppose in a proof there is a statement "Let $x$ be any arbitrary real number". Semantically, I understand that this means "Consider for any (fixed) real number and let us call that $x$ for the sake of referencing in the future". However, I am not quite sure how should I think about the referenced object.

  1. Should one vaguely think of some real number in mind such as $0.031$ or $3145$, or $-3$ (And after the reasoning about that particular object is completed, in one's mind, they argue that the reasoning can be applied to any real number)?

    Problems that occur to me: If one has some specific number in mind, it seems to me that it will increase the probability that the reasoning will not be general, and one has to be aware of a corner case to be confident that they arrive at the generality. To me, this seems that there is a loss in the rigor of reasoning which decreases the degree of belief in the proposition being proved.

  2. Should one remain at a highly abstract level, i.e., treats it as some mathematical object that is in $\mathbb{R}$ without thinking about any specific number?

    Problems that occur to me: If one thinks about it very abstractly, to me, it is really difficult to reason about this abstract object, a proof will be unintuitive, and it takes a relatively long time to understand a proof. To me, this seems more rigorous but at the sacrifice of the enjoyment of doing mathematics (The problem can be seen when reading "Let G be any graph" and not imagining any vertices and edges or reading "Let A be any matrix" and not imagining any specific dimension)

Or in the end, we cannot be sure of any proof we have done unless we prove them syntactically using rules in an axiom system like a game?

Do you have any insight/advice on how should one think/what should be going on in their brain regarding the issue? (Or do you have any recommendations of philosophy/logic books discussing the issue? I began reading Shoenfield's mathematical logic book a while ago but he states that the book studies logic syntactically, so I stopped since I am not sure whether the book will address this issue)

Thank you in advance!

Thanatach Tangsakul
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    I just think: $x$ is a real number and I don’t know anything else about it. I do know that the axioms for the real numbers are satisfied. I might picture the number line and remember that $x$ corresponds to one of the points on this line. I might think of some specific numbers like $2$ or $\pi$ or something and think things like “for example, $x$ might be $2$…” – littleO Jul 17 '22 at 17:11
  • The main thing to watch out for is that you don't use any unstated properties of $x$ (e.g. it would be wrong to start with "let $x$ be a real number" and then use some argument that only applies to integers). Temporarily thinking of $x$ as a concrete real number is a mental aid in checking this, but is admittedly not exhaustive. – angryavian Jul 17 '22 at 17:14
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    It’s often very helpful to think about some good concrete examples to help guide you through abstract reasoning. I think mathematicians do this all the time. – littleO Jul 17 '22 at 17:15
  • How much do you know about universal quantifiers, i.e. the "for all" quantifier? I ask because the expression you are asking about is just an English language expression that is used to describe the application of the logical rules regarding universal quantifiers. – Lee Mosher Jul 17 '22 at 17:18
  • Your question is reasonable, but just in case you aren't aware, "let $x$ be an arbitrary real number; then blah blah" ("any arbitrary" doesn't quite seem to properly capture the meaning, btw) can be reframed as "for each real $x,$ blah blah"; then, apply littleO's remarks above. – ryang Jul 17 '22 at 18:13
  • @ryang I'm quite aware that we have to add "for each/every". To me, there is another problem with how to think about this statement. This resides in my mind for a long time. Since I cannot iterate through every single number to apply an argument, I cannot explain whether I truly understand it. It seems as if I only partially understand the meaning of a "for all ..." statement and think about the statement via language without truly understanding it. The fact that I am not able to consciously visualize the true meaning of a statement/proof makes it hard to believe that I understand mathematics. – Thanatach Tangsakul Jul 18 '22 at 05:04
  • To my limited knowledge, I suppose we could translate the semantics into symbols and use rules of inference in syntactic logic to explain that the reasoning is correct. But, to me, that doesn't help explain that we understand mathematics at all. – Thanatach Tangsakul Jul 18 '22 at 05:05
  • @LeeMosher My current understanding is that it is used to state a proposition that semantically we want to interpret the symbol as "for all" of something. But I am not sure to what extent we can believe in the semantics (interpreted by humans) of the rules of inference and any true formula derived from those. – Thanatach Tangsakul Jul 18 '22 at 05:13
  • I personally avoid saying “for all $x \in \mathbb R$” and instead I say things like “if $x$ is a real number then such and such is true “. – littleO Jul 18 '22 at 05:13
  • @littleO I tend to do this too. But It feels like, in your head, imagining a sphere to be a set of real numbers, iterating through things (numbers) in them, and thinking that such and such is true seems to be more intimate to the truth (and more organized/intuitive when thinking about any specific number later); however, both ways seems to fall short in thinking/understanding the infinitely many numbers of truth. – Thanatach Tangsakul Jul 18 '22 at 05:26
  • See this post – Mauro ALLEGRANZA Jul 18 '22 at 07:48
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    [Re: 6th comment above this] 1. "To me, there is another problem with how to think about this statement." Isn't this the same dissatisfaction (which I said is reasonable) that is the thesis of the above post? $\quad$ 2. "I'm quite aware that we have to add 'for each/every'." No no, you're replacing, not adding, and the replacement is not compulsory. And since 'every' suggests that we are considering the entire collection of real numbers as a whole, here, 'each' is a cleaner, less distracting, and preferable framing. – ryang Jul 18 '22 at 07:50
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  • "I suppose we could translate the semantics into symbols and use rules of inference in syntactic logic" Not even: you'd also have to invoke the relevant axioms. But all this is unnecessarily formal; the dissatisfaction that you are raising is analogous to the issue of how one can be sure that every positive number is bigger than -3, given that it is impossible to iterate through every single positive number. To this, I second @littleO's suggestions about invoking a representative object, even as this suggestion sounds circular to your dissatisfaction, haha. No space to elaborate...
    • – ryang Jul 18 '22 at 07:56
  • @LeeMosher: Please see this post, which explains why it is a bad idea to continue using "let" in this problematic way, especially when teaching beginners in basic FOL reasoning. – user21820 Jul 18 '22 at 12:14
  • Maybe use "suppose (for the sake of argument) that..." instead of "let..." to emphasize the provisional nature of the statement following? Or "what if..." – Dan Christensen Jul 18 '22 at 13:48
  • Elaborating on my point#3 (and @littleO's suggestion) above: are you aware that "let $x$ be an arbitrary real number; then..." is actually short for "let $x$ be arbitrary such that $x$ is real; then..."? $\quad$ The 1st component is making a declaration, whereas the 2nd component is specifying a condition/hypothesis: $\forall x{\in}\mathbb R;P(x):\equiv:\forall x \big(x{\in}\mathbb R{\implies}P(x)\big).$ Hopefully this deconstruction is helpful. – ryang Jul 19 '22 at 06:14
  • @ThanatachTangsakul: Do you understand my answer? – user21820 Jul 26 '22 at 12:17
  • @user21820 I think I quite grasp the idea. However, putting this (the abstract thinking similar to the second case I outlined) into practice is quite exhausting, especially when there are many abstract objects and a proof involves a lot of algebraic operations. The main problem, still, is that it is hard to be confident that I understand the semantics of the proof's arguments when such abstract objects are involved. – Thanatach Tangsakul Jul 27 '22 at 15:11
  • To illustrate the point, when a proof references some object that doesn't exist but we suppose it exists (in a proof by contradiction), the difficulty comes from the fact that we cannot imagine any specific example in our mind because any specific object would not have such property, so to have one in mind seems wrong to me. But not having one, the proof is completely not intuitive to me. – Thanatach Tangsakul Jul 27 '22 at 15:12
  • (I also have a side question that I think it would be perfect to address here) Before I got into analysis, when I read number theory almost everything is just a symbol manipulation (both me and seemingly the author). After I became aware of the problem of semantical understanding, I decided to reread everything because what I had done so far is just a meaningless computation. However, my mental capacity runs out pretty quickly not manipulating any symbols. I'm not sure if this is the right way because my pacing is really slow. What do you think? – Thanatach Tangsakul Jul 27 '22 at 15:14
  • @ThanatachTangsakul: You're right that in a subproof by contradiction, we may not even have any example of an object referenced in that subproof. That's why you should get yourself familiar with formal proofs (especially in a Fitch-style system), because ultimately the only way you can have a complete grasp of mathematics is if you fully understand the FOL reasoning. No doubt, intuition plays an important role in finding proofs. But it cannot play a role in actually understanding how a theorem is true. – user21820 Jul 27 '22 at 15:15
  • Regarding number theory, it's true that discrete facts tend to be ad-hoc and not as 'intuitive' as continuous facts. But I don't think it is necesssary to always understand a proof "semantically". Game semantics works all the time, but it's not necessary either. (By the way, if you are satisfied with my answer, you can click the tick to accept it.) – user21820 Jul 27 '22 at 15:36
  • @user21820 I understand that what you say is the right way to understand whether a particular proof is valid. But could you elaborate a little more on why that is the way to understand mathematics? To my very limited knowledge, that way doesn't quite capture the essence of mathematics. (If you could tailor your elaboration specifically for platonist that would really great) Moreover, (in the fields of mathematics that currently I find beautiful,) I think most of the proofs are not formal proofs. Did you suggest that mathematicians in these fields do not understand why the proof is truly valid? – Thanatach Tangsakul Jul 27 '22 at 15:39
  • @user21820 Regarding the answer accepting, I really want to see various thoughts and opinions based on many other philosophies (I'm not sure whether accepting any answer would discourage any such action). Also, I am not sure whether I should judge to accept any answer based on my subjective thought since I ask for something that doesn't seem to have only one answer. – Thanatach Tangsakul Jul 27 '22 at 15:53
  • As far as modern mathematics is concerned, most of it is mere symbolic consequence of the chosen foundational system. Although full ZFC is not needed for most modern mathematics, the abstract fields require a significant part, roughly bounded ZC plus bounded replacement (i.e. "bounded" meaning restricted to bounded defining formulae). This has almost nothing to do with platonic reality, since as far as we know all applied mathematics can be done within an extremely weak system called ACA0, and all mathematics that has real-world meaning seems to lie roughly within ATR0. – user21820 Jul 27 '22 at 15:58
  • ATR0 is way below Z2 (second-order arithmetic), and Z2 is way below BZC. Look up reverse mathematics regarding ACA0 and ATR0 and Z2. Note that Z2 is about just ℕ and subsets of ℕ, but some logicians aren't even confident that Z2 has real-world meaning, since its comprehension principle is impredicative, and we have no clear evidence that impredicativity is meaningful. So if you want to talk about platonism, you're going to be restricted to roughly ATR0, not modern mathematics. – user21820 Jul 27 '22 at 16:02
  • About mathematicians who don't use formal proofs, they do understand FOL reasoning to some extent (whether consciously or not), but if they don't know formal proofs then they actually rely on their intuition as a key crutch in searching for mathematical arguments. Their intuition may of course be faulty. That's why I have found irreparable flaws in the arguments made by some professional mathematicians who taught me in my undergraduate courses, not just once or twice and not just by one person, and some of them take a long time to even understand that they are wrong. – user21820 Jul 27 '22 at 16:05
  • @user21820 Could you elaborate more on what do you mean by symbolic consequence? Because, for example, (to me) the beauty in topology or geometry can be seen semantically with an absence of symbols. (Especially in algorithms, I think one can see that the beauty/concept/truth is there even if one doesn't know anything about the foundation of mathematics or even any basic axioms.) I am confused because I rarely see a pure symbolic proof in a typical field. – Thanatach Tangsakul Jul 27 '22 at 16:13
  • @ThanatachTangsakul: Euclidean geometry is actually heavily axiom-based. In absence of a synthetic axiomatization, you would not have any beauty. Same in topology. Just because you can draw pictures doesn't mean that you actually have a solid understanding... For example, many of not most youtube videos on the Borsuk-Ulam theorem have WRONG poofs precisely because they made intuitive but bogus leaps of faith. – user21820 Jul 27 '22 at 16:19
  • Also, just because published mathematical articles are mostly informal or only semi-formal, doesn't mean much. If the mathematician who published it has gotten a rigorous FOL proof, the lack of formalization means nothing. But if not, then you can't be sure that it's not wrong. Witness how Kempe's wrong proof of the 4-colour theorem was accepted by others for a decade... There are recent examples as well, such as Nelson's wrong proof that PA is inconsistent... – user21820 Jul 27 '22 at 16:23
  • So are you still unsatisfied? – user21820 Aug 02 '22 at 19:18
  • @user21820 As I stated 8 comments above, I really want to see other thoughts that might give a better insight since thinking abstractly as I do now is pretty tedious. If there is any downside to this being left open, please let me know. – Thanatach Tangsakul Aug 03 '22 at 11:51
  • Well I asked because you didn't respond to my last rebuttal of your assumptions, which would undermine your notion that there is a 'shortcut'. What I'm saying is that if you want to be sure of doing correct mathematics, then systematic FOL reasoning is necessary, whether or not you use the same symbols. Up till now, it seems you haven't seriously studied what I brought up (e.g. Fitch-style, game semantics, reverse mathematics, Borsuk-Ulam, Kempe's wrong proof). I don't think you can grasp what is important if you don't deal with the real solid stuff. – user21820 Aug 03 '22 at 12:33
  • There's no downside to waiting for another answer, if you wish. But I think you need to actually spend lots of effort to learn the solid stuff otherwise whatever I'm talking about is simply inaccessible to you. – user21820 Aug 03 '22 at 12:35
  • @user21820 I did spend a lot of time trying to understand things you mentioned and, of course, there are a lot of things I couldn't understand at the moment (Of course, I plan to read them all in the future when the time is right). Again, I do not disagree with you that for us to be (non-philosophically) completely sure that the proof is right, it has to be a formal proof. But my question is concerned with how to rightfully understand the semantics of a proof's statements that are expressed in natural language because most textbooks write proofs in this way. – Thanatach Tangsakul Aug 03 '22 at 16:50
  • (The reason I want to understand things semantically is because when I think in terms of symbols and formal logic as I did in my introductory course to logic, I do not feel like I am doing mathematics at all. Of course, I am convinced that the proof is correct in this way. But It is much greater when I can understand the proof semantically since I personally feel like this is the right way to see the beauty of mathematics.) – Thanatach Tangsakul Aug 03 '22 at 16:57
  • Ah, I think your last two comments clarify your inquiry significantly. The answer is that often people do intuitive handwaving instead of 'solid' mathematics. This means that they come up with nice-sounding assumptions that they have not proven but would like to have to continue their reasoning, and just skip that. When doing so, they usually do think that their assumptions can be proven, just that they are too lazy to do it or it appears too tedious. Frequently, they are right. Because of that, what we see published omit those stuff, and hence we have a skewed picture of 'mathematics'. – user21820 Aug 03 '22 at 17:22
  • So whenever people say "mathematics", this skewed picture is what most people (clearly including yourself) have in their minds. This isn't 'wrong'; it's just that it's more accurate to call it "published mathematics". Also, there's a way to 'grasp' the "semantics" of the (natural language) intuitive handwaving, without sacrificing rigour. That way is to (of course) prove that there exists a model of the desired axiomatization. It will not be any easier. This way is how one can work entirely within synthetic geometry while knowing that it is not a bunch of inconsistency or meaninglessness. – user21820 Aug 03 '22 at 17:26
  • Perhaps this is what you were looking for by "see the beauty of mathematics". In particular, you can fully utilize a beautiful axiomatization once you prove that it holds for the structure you wish to analyze. If you feel that these last comments help, I can move them into my answer. – user21820 Aug 03 '22 at 17:51
  • @user21820 I think my last question is somewhat not directly related to the main question I posted here so I am not sure whether that should be done (I also have a difficult time trying to understand how your last responses are the answer to my last question). Anyway, I think we should end the discussion here because I did not intend to get an answer to that last question at the moment. Thank you so much for your response, it has been a great help! – Thanatach Tangsakul Aug 04 '22 at 11:15
  • @ThanatachTangsakul: Just to clarify, I was explaining why you "do not feel like [you are] doing mathematics at all [whenever you] think in terms of symbols and formal proofs". I'm saying that this is because what people (and you) think of as "mathematics" is more like "published mathematics", which for obvious reasons omits most of the 'tedious steps in reasoning'. For example, you can take a look under "Rigour + Intuition" and "Beautiful stuff" on my profile page; they all omit formal reasoning, but they are 'actual' mathematics only because they can be formalized. You're welcome! =) – user21820 Aug 04 '22 at 12:27