Does every proof need an axiom saying it works?

Question

I am wondering whether for every (valid) proof $P$ done in mathematics, at least one of the following statements are true:

1. There is an axiom guaranteeing that its schema indeed gives us license to conclude the truth of what it purports to prove (an axiom saying the proof works, like how the axiom of induction in the Peano axioms says an inductive proof works).

2. There is a finite sequence $P, P', P'', \ldots Q$, where the ($i+1$)st element of the sequence is a proof that the $i$th proof is correct, and there is an axiom saying that the schema in $Q$ works to prove what it is trying to prove.

The reason why I am asking was because I thought that when we prove something, we are just using common-sense rules innate to our minds; yet comments to my post here say to be careful using common-sense while proving things, and to instead use rigorous logical rules, leading me to wonder whether for any valid proof, at least one of those 2 things holds above.

Answer 1

This is a complicated question to answer for multiple reasons. We really have to say something about the foundational crisis to give a full account of the story here.

Here is a caricatured brief version of the story. Back in the old days (Euler and Gauss and so forth) mathematicians studied fairly concrete objects (specific numbers and functions and equations) and did so with an informal mix of intuition, proof, and calculation. This mostly worked fine. Then various people noticed, around the 19th century and early 20th century, that this approach could sometimes get you in trouble, and that it was less clear than they had thought which mathematical objects really existed and which mathematical statements were really true; examples include

• the discovery of non-Euclidean geometries, showing that the parallel postulate was independent of the other axioms of Euclidean geometry,
• counterexamples to a naive form of the Dirichlet principle,
• the discovery of the Weierstrass function, a continuous function which is nowhere differentiable (people used to believe that continuous functions were "differentiable almost everywhere"),

and probably other important examples I've forgotten. So eventually mathematicians decided they had to get a lot more careful about how they did mathematics. People wanted some kind of rigorous foundation from which they could deduce the rest of mathematics and prove it was all consistent and so forth. Frege tried do do this but Russell showed that his system suffered from Russell's paradox.

Hilbert proposed that all of mathematics should be written in a precise formal language with precise rules and axioms describing what proofs are valid, and then it should be proven that this whole system was consistent (did not prove any false mathematical statements) and complete (proved all true mathematical statements). But Godel showed that this was impossible in an extremely strong sense. Oh no!

(For a more detailed and quite lovely version of this story you can check out Logicomix.)

Nevertheless, out of this whole struggle emerged foundations for mathematics that work, more or less. Those foundations are called ZFC set theory, and what they do is provide a last-ditch option that a mathematician is supposed to be able to use to convince another mathematician that some mathematical statement is really true or that some mathematical object really exists: exhibit a proof or a construction in ZFC set theory. ZFC is widely believed to be consistent (although, by Godel, ZFC cannot itself prove this fact) but is not complete (for example it neither proves nor disproves the continuum hypothesis).

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Because of the foundational crisis, part of what you are supposed to do in mathematical training, at some point, is learn how to construct mathematical objects and prove things about them in set theory. The most important example is probably learning how to construct the real numbers in set theory, via Dedekind cuts or Cauchy sequences. Then you can prove, rigorously, using this construction, standard properties of the real numbers such as the least upper bound property. The idea here is supposed to be that, in theory, you are supposed to learn how to rigorously reason about mathematics in such a way that all of your arguments could, in theory, be expanded all the way until they were proofs in ZFC set theory from the ZFC axioms.

However, in practice we never actually perform this expansion. ZFC is like a very low-level programming language for mathematics, analogous to assembly language. What mathematicians actually do in practice is reason in a much higher-level language (analogous to, say, Python), which nobody ever formally specifies but which essentially everyone agrees on, and which you are supposed to pick up by watching other people use it. This high-level language is supposed to "compile down" to ZFC set theory but I don't believe anyone ever actually checks this in detail. The high-level language is much closer to type theory than to set theory: for example it contains a type for the integers $\mathbb{Z}$, a type for the reals $\mathbb{R}$, and so forth, and in practice we essentially never refer to their constructions in set theory, unless we are talking to a student who is working through those constructions.

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Okay, so now we can actually address the question:

What is a proof?

There is a certain myth we teach to students, which is that a proof is a rigorous sequence of deductions from axioms. I believe that much of the reason we teach this myth to students is the foundational crisis, and the enormous change it provoked in how people thought and talked about mathematics. This myth is part of how we transition students to what Terence Tao calls the rigorous stage of mathematical education, and it is often accompanied by ghost stories of the form "ooh, look, this intuitively appealing conjecture is false! Watch out!"

But the myth is not true! If you read any mathematical paper on nearly any subject other than set theory you will not find a single ZFC axiom anywhere. This is not how anyone actually does mathematics, because it is way too unbelievably tedious, and way too disconnected from intuition.

What is completely missing from the proof myth is that mathematics is fundamentally a social activity, and proofs are fundamentally a form of communication. A proof is something you write for other people to read and get something out of. It is a form of literature. Like any form of literature, it has genres and fashions and trends, and in particular standards for what constitutes a valid proof (in the same way that we can have standards for what constitutes a valid novel, say), and these standards are fundamentally something that specific communities of people actively negotiate amongst themselves, and which change. And also, like any form of literature, proofs can be beautiful, inspiring, delightful...

If you go to graduate school in mathematics and start specializing in a subject you'll read books and papers in that subject and learn what types of arguments are common in that subject; you become confident in those arguments not by reducing them all the way down to ZFC but through your confidence in the community of mathematicians in your subject, and through your personal working through the logic and the applications of those arguments. This involves a complex, delicate, and highly personal interplay between intuition and proof (and hopefully lots of calculations too!), and you need all of these ingredients working together; this is what Terence Tao calls the post-rigorous stage of mathematical education.

So: it's complicated! As you study mathematics more your conception of what a proof is will itself change, and this is completely natural. Good luck!

Answer 2

I will expand on my comment with an example. I think this will answer your question.

Every proof is, at its core, a derivation from axioms and deduction rules. Deduction rules are rules for proving new true statements (called theorems) from old ones. Generally, our axioms and deduction rules are encompassed in $\textsf{ZFC}$ and first-order logic.

Let us do a simple example (paraphrased from Gödel Escher Bach by Hofstadter, his $\textsf{MU}$ formal system).

Let us start with the axiom: $$\textsf{MI}$$

This is just a string of symbols. Now we present deduction rules for generating new strings of symbols.

1. $$x\textsf{I} \implies x\textsf{IU}$$
2. $$\textsf{M}x \implies \textsf{M}xx$$
3. $$x\textsf{III}y \implies x\textsf{U}y$$
4. $$x\textsf{UU}y \implies xy$$

Where $x,y$ are variables representing strings of symbols.

Now, what can you prove from this formal system? Maybe play around with it. Here’s an example of what I mean:

$$\textsf{MI} \implies \textsf{MIU}$$

by the first deduction rule. Since $\textsf{MI}$ is an axiom, we have proved $\textsf{MIU}$.

Now we can recursively (repeatedly) apply our deduction rules to this new theorem! For example,

$$\textsf{MIU}\implies \textsf{MIUIU}$$

by the second deduction rule.

And so on. The puzzle Hofstadter gives is, prove $\textsf{MU}$ or prove that $\textsf{MU}$ is not a theorem of this formal system.

Notice that this second statement is a meta-statement, which requires stepping outside the formal system itself, and into something else. It is a good puzzle, I would think about it.

Nevertheless, this a formal system just like $\textsf{ZFC}$, if less complicated. Every statement of math is a string of symbols, and if those symbols could be reached by applying deduction rules repeatedly to axioms, then that statement is a theorem.

Usually, we are working in some area, with some theorems already in our toolbelt, and can prove new things from those theorems, and not directly from axioms. This is what I mean by “recursively applying”. The deduction rules we use should be familiar to you if you’ve taken a logic and proof class (think modus ponens).

Does this answer your question?

Answer 3

In your question you say you are asking because you

thought that when we prove something, we are just using common-sense rules innate to our minds;

Most of the time that is in fact the case: this is part of @QiaochuYuan 's lovely answer, reproduced here with his permission.

There is a certain myth we teach to students, which is that a proof is a rigorous sequence of deductions from axioms. ... But the myth is not true! If you read any mathematical paper on nearly any subject other than set theory you will not find a single ZFC axiom anywhere. This is not how anyone actually does mathematics, because it is way too unbelievably tedious, and way too disconnected from intuition.

What is completely missing from the proof myth is that mathematics is fundamentally a social activity, and proofs are fundamentally a form of communication. A proof is something you write for other people to read and get something out of. It is a form of literature. Like any form of literature, it has genres and fashions and trends, and in particular standards for what constitutes a valid proof (in the same way that we can have standards for what constitutes a valid novel, say), and these standards are fundamentally something that specific communities of people actively negotiate amongst themselves, and which change. And also, like any form of literature, proofs can be beautiful, inspiring, delightful...

When I teach discrete mathematics (essentially the first course that calls for "proofs") I tell my students that a good proof on an assignment convinces me that they have convinced themself for good reason.

You are reading Tao, which is not a text for a first course calling for proof. Early in that text what counts as "good reasons" calls for much more formal arguments. Later the arguments will look more like "using common-sense rules innate to our minds".