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In modeling any real physical system, we are required to employ inductive reasoning. We can never be completely certain about the state or properties of any system or of any future observation we will make of it. All we can do is attempt to ascertain its properties from past observations, which are themselves imprecise approximations to the true complexities of the system.

It is a law of physics that at the quantum level, no information is completely certain. Everything that is observable exists within a certain probability interval, and there are fundamental limits to how much one can know about any physical state.

Mathematics generally employs deductive reasoning. We postulate that there are certain axioms which are absolutely true, and draw further inferences from those axioms. This is a very different type of reasoning from what I discussed above, as no such certainties exist in nature.

Given these differences, how is it then that our minds are so easily able to conceptualize mathematical certainties, given how contrary to nature they are? If our minds are themselves a part of nature, and have evolved to model and observe natural systems, what is it that gives us the capability to even conceive of things which have so much more precision than we would ever actually see?

Indeed, it is generally far simpler and easier to describe model a mathematical certainty than anything as complex and imprecise as a real system. In modeling physical systems, we generally employ “simplified” models that rely on assumptions about the properties of the system being much more precise and well-defined mathematically than they really are. Why should it be easier for us to, as physical being within this universe, model a physical system in such an aphysical way?

It would be quite simple for me to give a mathematical description of a perfect circle, and we could all, as rational beings, quickly agree on its properties. Yet, if I were to try and construct a circle, anything I constructed would not only be only an approximation to the mathematical ideal, but also much larger and more cumbersome than the mathematical description was. What is it about mathematics that makes it so much easier to communicate and reason about than nature itself?

  • My opinion: mathematics is purely a system of symbolic manipulation that exists only in our minds. The fact that it applies to reality is sort of nice, but mathematics doesn't need reality to exist: it's a purely rule-based system. – barrycarter Sep 2 '18 at 20:52
  • I do not quite follow the puzzlement. Why should there be any correlation between processing simplicity and occurrence in nature? Nature is complex, schematic rules are not, that is what makes them easy to process. Evolution would select for the simplest possible schemes that come close enough to survive. – Conifold Sep 2 '18 at 21:03
  • I'm not sure if you can call a mathematical concept precise. I think precision is more of a measure how well a concept matches reality or how well a measurement translates reality to some conceptual world, but saying mathematical things are of higher precision than things we actually see seems difficult to me. – fweth Sep 2 '18 at 21:27
  • @fweth: You seem to be describing accuracy, not precision. – user6559 Sep 3 '18 at 1:51
  • @Hurkyl yeah, you're right. Just one question: there are lots of phenomena in the world which occur pretty regularly, however, the irregularities pose very intricate and complex patterns. Since the brain tries to predict the world using it's limited computational resources, it would seem reasonable that it adopts a mathematical model which only implements the regularities in the given data. Would you also consider such a model as more precise than reality? I'm not so sure about it, it seems first and foremost simpler or more effectively but more precise? – fweth Sep 3 '18 at 15:49
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Just as in religion there is a leap of faith then in science there is a leap of understanding.

The essential notion that physical theory relies upon is that of a universal order; given that understanding, induction are merely attempts to work out the nature, detail and relationships of this universal order.

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Pierre Duhem, in his Aim & Structure of Physical Theory pt. 2 ch. 3 "Mathematical Deduction & Physical Theory" is worth reading. §3 gives "an example of a mathematical deduction that can never be utilized" in a physical theory, which is quoted in ch. 5 of the free Chaos film.* In other words: There are mathematical deductions that do not correspond to anything in the physical world.

*The entire film is worth watching, as its other chapters relate to your question, too.

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Your question includes a misconception about how we learn about the physical world. Induction is supposedly a process that allows us to use observations to arrive at theories and show that they are true, or probably true or good or something like that. But theories are accounts of how and why something happens and no finite set of observations is equivalent to such an account, so there is no way to get a theory from observations.

We actually create knowledge by noticing problems with our current ideas, guessing solutions to those problems and then criticising the guesses until only one is left and it has no known criticisms. We also learn about maths in the same way - by guessing and criticism.

To learn about mathematical objects we construct physical systems that model those objects according to our best guess about the physical systems and the mathematical objects. In some respects those models and the abstractions they model won't match perfectly. The mathematical objects we are good at modelling are those for which it is possible to construct arbitrarily accurate simulations. For example, you can construct an arbitrarily accurate simulation of a circle by using string of a particular width and making the circle larger to satisfy whatever constraint you want to put on the width of the border relative to the size of the circle. But there is a large subset of mathematics we can't model because no physical system can model them, these include uncomputable functions.

If you want to understand the issues involved better see "Proofs and Refutations" by Lakatos and "The Fabric of Reality" by David Deutsch, chapters 3-7,10. You can discuss them here http://fallibleideas.com/discussion.

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