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This is probably one of the biggest question I have when learning some mathematics. I always wonder if I have a concept in my head lets say continuity. Lets I want this concept to be able to characterize a function that has "no breaks"; it is completely connected. So lets say I then create the a definition of continuity just as the limit definition of continuity is. And then I make theorems, lemmas, etc off of that definition. So after I am done what I learned from these theorems I can now see is implied to any function I have that has no breaks.

My question is how am I to be sure whenever I make definition that it matches up with the concept I am thinking. Like in the example above if for some reason my definition didn't imply my concept of continuity I was trying to form the new knowledge I have from my proofs would be falsely applied to concepts that it wasn't actually talking.

This goes to my overall problem with how am I to be sure the mathematical definition match up the intuition/concepts I am thinking of in reality. One way I thought to reassure my self if mathematical definition matches up with reality is think of attributes of my idea as prove my definition implies these attributes. Also I am not saying that for my example that every continuous function has to be thought as one with no breaks but at the very least it must imply that.

This also becomes more critical to me when the definitions become less intuitive and more abstract

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You don't. It's like some kind of axiom: You have an intuition about what a continuous function is. Someone goes and defines it in another way that makes it much easier to prove things about it. Only it's up to you to verify, intuitively, that the definition is equivalent to the intuition you have about a continuous function. –  Deathkamp Drone Jun 29 at 1:09
    
So you think in broad scope when when we say math implies a concept we have of reality it can only be mearly said to be true by pure intuition –  Kamster Jun 29 at 1:12
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If you can't define what a continuous function is, mathematically, then you don't really know what a continuous function is, that's the thing. For example, what is something that has "no breaks" for you? Some people would say to you that the rational line has no breaks, while others would it has breaks and that the real line doesn't. It's a problem because you never defined what is a "break" in first place. Intuition has its problems. –  Deathkamp Drone Jun 29 at 1:24
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Defining things mathematically isn't that simple so you shouldn't worry. Ancient mathematicians didn't define what they were studying precisely, yet they arrived at true and non-trivial results by means of intuitive proofs. Today mathematicians use a different approach, that is more abstract and rigorous. If you're going to do modern mathematics, then you have to be comfortable with the fact that sometimes you have to give up intuition and think abstractly and definition-wise, and that many mathematical concepts have no real life analogies. –  Deathkamp Drone Jun 29 at 1:38
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You might think that by now we'd have a pretty good idea of what the real numbers are, but we can't even tell whether there is a subset too big to be put into one-one correspondence with the integers, but too small to be put into one-one correspondence with the reals. So as @Deathkamp says, defining things mathematically isn't that simple. –  Gerry Myerson Jun 29 at 1:41

3 Answers 3

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The point of mathematics is not to match to reality.

This might have been the one of the origins of mathematics, but it has grown and evolved long beyond that point. Mathematics is its own universe, and it deals with assumptions and consequences.

How do you know that mathematical definitions match reality? You don't. If you are in fact trying to model real world settings, then you start with one set of definitions, and you try some test cases. If your definitions fit, then you continue, if not then you change them. And the process continues on and on.

But mathematics, in general, is no longer dealing with modeling reality. If it were, there were no infinite sets; no infinite objects; nothing larger than $2^{100000000000}$; no fractions; no irrational numbers; no negative numbers; absolutely nothing but $1,2,3,\ldots,n$ for some $n<2^{100000000000}$.

And besides, how do you know that your perception of reality is a good model of the actual reality?

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I guess one way I think whether math is true of reality I stay to the way Kant describes how math is true of reality (where the only reality Im concerned about is the observed reality) in that it describes the forms of our sensibility, time and space. But i see as you where saying that much math doesn't delve into these kinds of concepts anymore –  Kamster Jun 29 at 2:24
    
I like the point you made that mathematics models reality, but does not perfectly match reality. Mathematics is purposely an abstraction, a perfect world with real world benefits. –  qwr Jun 29 at 2:25
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@user159813: Whatever Kant said needs to be taken with a grain of salt regarding mathematics that was developed centuries after his death. –  Asaf Karagila Jun 29 at 2:30
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@qwr: No, my point is that mathematics doesn't model reality at all. We sometimes force this onto mathematics, but this will never be an accurate modeling for the same reason that an actual C++ program cannot really compute the entire set of prime numbers. –  Asaf Karagila Jun 29 at 2:31

This is not meant to be authoritative in any way, but it is my experience thus far. When learning math you often come in with intuitive concepts (such as continuity) and are taught the way that mathematicians have agreed to formalize those concepts (i.e. limits). This doesn't usually appear completely intuitive, and certainly the time it took to make Calculus more rigorous shows that it was not immediately apparent how to create the formal definitions which adequately captured the intuitive ideas. When learning existing concepts and definitions which are nearly universally accepted, I usually play with examples to see how the definitions best capture the intuition, and strive to find the counterexamples which delimit the usefulness of that definition's scope.

You mentioned reality in your question, but went on to give an example which many people may tell you has less reality to it than one would think - most physical things are not truly continuous, although it is often simplest to model them that way. That being said, I would like to split the remainder of my answer into a paragraph which discusses capturing "reality" as Mathematician's view it (i.e. reality of ideas and concepts currently undefined) and "reality" as most people would see it (i.e. physical reality).

When creating new definitions to encompass a concept, Mathematicians usually work with the concept extensively until they do begin to (hopefully) develop some intuition about it. At this point, many Mathematicians may have varying viewpoints on what aspects are important, and many proposed definitions are independently put forth. Indeed, one does not have to venture far from continuity to see other examples of definitions which may have well taken the place of importance we currently place on continuity, for instance the page on the Hölder condition contains a nice chain of inclusions for varying degrees of "smoothness" that we could expect for a function. Part of what is at play here is a sort of evolutionary aspect, where useful definitions are reused more frequently and eventually become universal. It is a delicate balance between generality and usefulness of the definition which eventually becomes the deciding factor.

When considering whether Mathematics accurately models "reality" in a physical sense, the question becomes more suited for other disciplines, but there is still certainly a significant question there. I was personally impressed by the level of detail given to this topic in Feynman's Lectures on Physics. Feynman introduces essentially the mathematical definition of a vector in physical terms, and then whenever introducing a new physical term which is a vector (i.e. velocity) he goes into great detail to show that it indeed does obey the axioms expected of a vector. (Read this online here.)

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"This also becomes more critical to me when the definitions become less intuitive and more abstract."

I think a sort of platonic reality must be the litmus test for definitions of more abstract phenomena which are not observable physics. For instance, I question if cardinality has been defined properly within the context of ZF when we are unable to decide whether $2^\omega <2^{\omega_1}$ without basically adding it as an axiom. That is, we are unable to decide whether a larger set has more subsets. Absurd.

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