On "why" questions in mathematics

Question

In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains:

In a more general setting, one needs to remember that $0,1,2,3,…$ are just symbols. They are devoid of meaning until we give them such, and when we write $1$ we often think of the multiplicative identity. However, as I wrote in the first part, this is often dependent on the axioms - our "ground rules".

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This is why the question "Why $1+1=2$?" is nearly meaningless - since you don't have a formal framework, and the interpretation (while assumed to be the natural one) is ill-defined.

Questions:

1. To what extent is it true of all "why" questions in math that they are "nearly meaningless"?
2. Is the answer to question #1 different depending on which facts you're asking "why" about? -- are there certain types of statements that can have a meaningful "because" and others that can't? For example, is there a fundamental difference in our ability to pose and answer "why" questions between statements immediately defined to be true, and those that follow from other things?

Answer 1

I believe it is not true to any extent that all "why" questions in mathematics are nearly meaningless.

What Asaf describes is, that it is pointless to ask "Why $A$ holds" if you have not even defined the symbols / terms that occur in $A$ and if you have not fixed an underlying logic, that is rules of inference and axioms.

If you have, then it is perfectly okay to ask "Why does $A$ hold", which could mean that you ask for (among other things):

1. literally a proof of $A$
2. a proof sketch of $A$
3. a key property used in a proof of $A$ that the objects in question have, that other objects do not have ("Why do reals form a field but real $2\times 2$ matrices don't?")

Questions in mathematics do not need to be ultra precise all the time, after all mathematics is still usually done by people.

Answer 2

In an absolutely formal sense, any "why" question in mathematics requires an axiom system. But most proofs are not absolutely formal.

When dealing with proofs at the very lowest level, you really need to know the axiom system you are using to know how to prove something. It might just be the definition of $2$ that $2=1+1$. So it is meaningless to ask how to prove it without knowing the axiom system.

On the other hand, there are lots of equivalent axiom systems for number theory. And often when we are asking for a proof of a higher-level theorem, you might not care which axiom system you were using. You reference statements that are known to be true in all the equivalent axiom systems, and aren't really concerned about what is an axiom and what is proved.

However, if you were asked to write a 100% formal explanation of "why?" for any theorem, you'd need to know the syntax and axioms in use.

But we rarely want 100% formal proofs.

Consider it the difference between "Describe an algorithm to compute $X$" and "Write a computer program to compute $X$." The computer program depends on the machine and language. The algorithm is a more general idea, which simply assumes broadly that certain types of computation can be done on the machine.

Answer 3

To expand on Stefan Perko's answer, here are three meaningful "why" questions in mathematics:

1. Why is Langrange's thorem true? I.e., if $G$ is a finite group and $x$ is an element of $G$, why does the order of $x$ divide the order of $G$?

2. Why are the "elementary proofs" of the prime number theorem much more difficult than proofs that use complex analysis?

3. Why has Goldbach's conjecture not been proved?

To answer, 1 we can give a well-known proof. To answer 2, we would have to do some metatheory and compare known proofs. To answer 3, we would have to find some way of reasoning about our capabilities for finding proofs or counter-examples.