I'm currently learning about probability theory, and a sudden question popped into my mind: how does math work. To me it seems that math consists of a bunch of symbols, such as $x$, $y$, $\theta$, $1$, $2$ etc., and a bunch of rules for manipulating those symbols, such as the production rule and summation rule. Using those components as building blocks, people are creating more rules that can either make our life easier or reveal more insights into a problem, like the Bayes theorem for example.

But this way of thinking definitely cannot model all the components of math. For instance, how could I incorporate the definitions like variance, expectation and Gaussian distribution etc. into this framework, and how to account for the proofs which play a huge role in math.

So, my question is, is there any conceptual framework that can formally and comprehensively model how math works? Thanks!


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  • $\begingroup$ Study abstract algebra and the Peano axioms. That would be my suggestion. Others might have better ones. $\endgroup$ – Clarinetist May 3 '18 at 20:11
  • $\begingroup$ Mathematics grew out of the necessity of measuring things and their relationships. In that sense, it is the science of measurement. $\endgroup$ – John Wayland Bales May 3 '18 at 20:16
  • $\begingroup$ Formalism was a (meta-)mathematical philosophy which asked a bit of the same questions. Can we use symbols to capture and express all of mathematics? $\endgroup$ – mathreadler May 3 '18 at 20:17
  • $\begingroup$ IMHO, since I am not a mathematician myself, you gave a possible description of how math works on the first paragraph, and I cannot see why the second paragraph would invalidate the first one. Of course, the development from building blocks is usually guided to a final result that the mathematician have already in mind, which is informed by their intuition and imagination. $\endgroup$ – toliveira May 3 '18 at 20:18
  • $\begingroup$ Formalism and axiomatic system seem quite interesting. I'll take a look! $\endgroup$ – NoSegfault May 3 '18 at 20:24

Mathematics is the art of recognizing numerical patterns in the world around us. While this is certainly a debate in the philosophy of mathematics, I am firmly in the "math is discovered" rather than "math is created" camp. Pattern recognition is at the heart of mathematics, however, and to do that requires imagination. Sure, there are formal rules about symbols and their manipulation - whole branches of mathematics (such as algebra) concern themselves with just that. But there's more to mathematics than symbol manipulation. It's an art form (see King's book The Art of Mathematics for an extensive discussion of this) more like poetry than anything else. Mathematicians do mathematics because it is beautiful.

How does mathematics work? Fundamentally, there is pattern recognition. Those patterns build into logical structures in which the mathematician's mind has to move. Frenkel's book Love and Math goes into this quite a bit.

This way of thinking about mathematics makes all of statistics absolutely indispensable. How could you make sense of data from the real world without statistics? This, by the way, shows that inductive reasoning (statistical reasoning) is just as important to the mathematical world as deductive reasoning (definition, theorem, proof). We do our students a great disservice if they think mathematics gets discovered (there's my bias, again) in definition-theorem-proof format.

  • $\begingroup$ I agree with you that a large amount of math knowledge comes from our experience with the world, and we can infer the logical rules based on statistics. But how could those more abstract concepts, such as lambda calculus be statistically derived? $\endgroup$ – NoSegfault May 3 '18 at 21:07
  • $\begingroup$ I guess here we are looking for a statistical system that could model math, which consists of both invented and discovered item? $\endgroup$ – NoSegfault May 3 '18 at 21:21
  • $\begingroup$ @StevenLi: I didn't say the more abstract concepts are necessarily derived from statistics. They can come from a wide variety of sources, and a wide variety of patterns. If you look at the history of mathematics, you will see a gradual increase in abstraction: patterns built on top of other patterns. If you want a system that could model math, the best claim to that would be character theory; some people say it's so abstract that it doesn't say anything useful. I'm not so sure of that, but I'm not category theory expert. $\endgroup$ – Adrian Keister May 4 '18 at 0:11
  • $\begingroup$ Another shot at it is the Langlands program, but I'm not sure Langlands aims at the unification of ALL mathematics. Certainly it's trying to unify a number of large branches. $\endgroup$ – Adrian Keister May 4 '18 at 0:12

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