Is math built on assumptions?
I just came across this statement when I was lecturing a student on math and strictly speaking I used:
Assuming that the value of $x$ equals <something>, ...
One of my students just rose and asked me:
Why do we assume so much in math? Is math really built on assumptions?
I couldn't answer him, for, as far as I know, a lot of things are just assumptions. For example:
- $1/\infty$ equals zero,
- $\sqrt{-1}$ is $i$, etc.
So would you guys mind telling whether math is built on assumptions?
To respond to the charge that "we assume so much in math": math involves the analysis of various hypotheticals. When I say "if X then Y," I might need to assume X hypothetically while in the process of proving the implication, but this does not require presumptuousness in any way. One may easily accept "if X then Y" and its proof without actually accepting X at all!
On the other hand, there are various configurations of axioms that can be chosen as the logical foundation of mathematics, with ZFC as the tacit standard. Outside of foundations and set theory and model theory and logic and so on, the axiom list is relatively small (compared to the whole of the theory of any mathematical context with all of the various theorems and formula) and more importantly unchanging, so this is not relevant to the charge "assuming so much in math."
And further away from presumptuousness, many things in math (as anywhere else) are not actually "assumptions" but are conventions and definitions. I don't "assume" an apple is a red or green fruit that grows on trees, for example, that's just how it's defined.
The basic assumption of the "working mathematician" is the following: The logical and set-theoretical environment of his deliberations is not contradictory.
When we say: "Assume the triangle $ABC$ has a right angle at $C$" then this is not an assumption about the "universe of truth", but the announcement that we are talking about right-angled triangles in the sequel.
In my very first class in the university, Linear Algebra I, the professor began by telling us the following thing which I have engraved deep into my memory:
Mathematics is a science of deductions. We assume certain things, and we infer conclusions from them.
Later I had learned that we also assume that our inference rules are sound, and that the foundations are solid. Of course we can't really prove these things in full, as that would amount to verifying infinitely many things, which we can't do. We always have to assume some consistency of the system we work with. Of course, we don't just randomly assemble rules and assume that they work, we have more than a few centuries where mathematics was based on "natural deductions", so to speak, that allows us to try and distil the assumptions we want to work with.
This is one of the arguments against mathematicians that formalists, or people who see mathematics as a formalism, take: it's just pushing symbols on a piece of papers, and it's all just vacuous because it's a lot of assumptions which rarely have something to do with reality.
But then again, can we really prove that yesterday happened? that we are alive? that we exist? Not in the full, mathematically rigorous, and fully convincing sense of the word. But we have some coherence, some consistency, and we are easily convinced of our own existence, and that it did not began yesterday. Even if it's all lies, and you're all in my head -- you wouldn't believe me if I told you that.
I think a better way to say this is that nothing is assumed in math. Nearly every mathematical statement is really saying: "if this is true, then this is also true".
Certain assumptions are so common that they are usually left off (suppose that $\mathbb{R}$ is a set of elements which has the following properties... ), but you should never make the mistake that anything is assumed by definition.
$\frac{1}{\infty} = 0$ is not an assumption, it is a convention. $\sqrt{-1} = i$ is just the symbolic representation. 'i' just represents $\sqrt{-1}$. When we start proving a theorem, we assume required hypothesis and try to get some new results by logical steps, so we assume something as a hypothesis in every theorem.