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First let me point out that I realise this is a subjective question, and so it doesn't follow the general guidelines for "a good question on MSE". However, I have read through the guidelines for "what makes a subjective question good" and I believe this question fits the mould.

"What is the biggest distinction in all of mathematics?"

In the abstract for a public lecture which will be occurring at my university soon, the speaker claimed that "the difference between discrete and continuous is the biggest distinction in all of maths". I believe this statement is meaningless, because they aren't mutually exclusive. For example, any function mapping between discrete spaces is automatically continuous.

I believe what the speaker was essentially trying to say was, "the biggest distinction in maths is whether a space is discrete or complete (in the metric sense)". But now if you try to capture the essence of the statement by abstracting it to a topological setting, it's like saying "the biggest distinction in maths is the discrete topology vs the trivial topology", which seems nonsensical. Almost nobody uses those topologies. Besides, there are so many other fields of maths, like logic or group theory or graph theory that barely mention topology at all.

My question is, is there a reasonable answer to the question "what is the biggest distinction in all of mathematics?" If so, what could it be, and why does it fit that description? If not, why not?

(I'm leaning towards "there is no answer to the question", but my only reasoning is that "maths is simply just too broad", which isn't a strong enough argument.)

Harambe
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    The finite and the infinite? – user4894 Nov 02 '17 at 00:55
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    Tangential comment, but this reminds me of a quote: "The great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity." -- R. T. Rockafellar – littleO Nov 02 '17 at 01:03
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    My first thought when I saw the title was to reply with "The Fields medal." The intended question is more interesting. – Mr. Chip Nov 02 '17 at 01:16
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    The author said: "the difference between discrete and continuous...". You said: "I believe... For example, any function mapping between discrete spaces is automatically continuous". The problem with your analysis is that (in my opinion) the word "continuous" does not have the same meaning in both sentences. I believe that the author is not talking about topological concepts. I suspect he refers to the nature of what is countable and what is uncountable. Maybe, this is how a statistician would interpret that sentence. See this. – Pedro Nov 02 '17 at 01:18
  • If I'm not wrong in my previous comment, he is talking about these senses of discrete and continuous, in which case he is probably right. – Pedro Nov 02 '17 at 01:27
  • Right now it's very obvious why this question has been put on hold because the only way I can "accept an answer" is "which one I liked most" in a purely opinionated way haha. I'll go back to asking/answering actual maths questions. – Harambe Nov 02 '17 at 06:34

3 Answers3

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The distinction between true and false.

Then again, maybe that is not the biggest distinction.

Bram28
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    I suspect the distinction between proved and not proved might be bigger. This seems to get more at the heart of mathematical practice. – Dave L. Renfro Nov 02 '17 at 01:03
  • @DaveL.Renfro Yeah, that one crossed my mind as well, but I picked this one, as I can take the claim that something is proven, and consider it true or false. Or: either my claim that the biggest distinction in math is between true and false is true, or it is false. – Bram28 Nov 02 '17 at 01:15
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    What about the undecidability, for example of the hypothesis of the continuum? – Piquito Nov 02 '17 at 01:19
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“God made integers, all else is the work of man” is a known Kronecker quote. Un answer could be precisely in relation to the numbers. In antiquity a good answer was manifestly the distinction between rational and irrational but closer in time could be the distinction between algebraic and transcendental. I think other equally relevant answers could be given.

Piquito
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If there is a grading allowed, then the following distinctions referred to by Lurie in this video were pretty convincing to me. They concern different levels of abstractions:

0: Numbers ($2=3$)

1: Mathematical Structures (groups, sets, etc.) ($k \cong k[x]/(x-1)$)

2: Categories ($V \cong_{nat} V^{**}$)

$\vdots$

The reason I find these distinctions interesting from a mathematical perspective is that the ``flavor" of argument is very different for when we decide (or prove) that two things are equivalent.

Andres Mejia
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