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Using mathematics a lot of 'laws' of nature are discovered, like the Higgsboson gravitational waves Diracs anti-matter etc. But if something like mathematics can predict how nature or reality 'should be' and is what does it say about the nature and reality. I sometimes think that if mathematics is usefull than the reality is perhaps symmetric and in balance or something like that. But is maths just putting a model on nature and you will find what you are looking for with the instruments and maths you use or does it really say something about nature?

  • "All models are wrong; some are useful." – Cort Ammon Mar 5 '16 at 5:29
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Mathematics never predicts how nature is, only how it might be. As @JoWehler pointed out, physics then has to select from the ways it might be the ones that have anything to do with what it appears to be, and refine those into approximations of what it is.

One way of looking at mathematics is as the combination of basic shared human logic with itself in all forms, as completely as possible. In that sense, mathematics contains the set of all clear descriptions consistent with those human understandings that are reliably shared between individuals.

If that is, in fact, what mathematics is, it should not surprise us that when we dream up systems of possible physics and communicate them, they are composed of clear concepts we can reliably share. We have no choice but to base science on mathematics, as otherwise it would be either idiosyncratic, or impossible to communicate beyond a certain audience.

So from that perspective, the fact we find mathematics everywhere has nothing to do with nature, and everything to do with us, instead.

  • +1 for your interesting thesis about mathematics, as expressed in the last paragraph. What about converting it into a separate question on this site? – Jo Wehler Mar 5 '16 at 2:24
  • A question about 'the unexpected efficacy of mathematics' or 'what exactly is mathematics' would be duplicate of many others already asked. I guess I could ask 'why is mathematics not psychology', but it would be a 'Pushing a personal philosophy' question. I could hide behind Brouwer, who said the same thing less clearly, but he is actually rather widely mocked. I don't see a version that gets taken seriously and doesn't get closed.... – jobermark Mar 7 '16 at 0:52
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It's the aim of physics not of mathematics to find out how nature works. Physicists design theories to explain the phenomena in certain domains of nature. There are mechanics, electrodynamics, solid body physics, quantum mechanics etc. These models are expressed in a mathematical language. Now mathematics can solve certain equations from these models and derive results which translate back into observable physical predictions.

The base is the physical model, not the mathematics.

Nevertheless it is a big question why the language of mathematics fits at describing nature, see the famous essay by Eugene Wigner https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

One possible, but a bit speculative explanation would be that our world is a mathematical simulation. This possibility has been brought up in the last time. But it has been criticized, because a digital simulation would imply that we observe anomalies due to the rounding errors of the digital computers used in simulation. For the whole issue see Chap. 10. Universes, Computers, and Mathematical Reality in Greene, Brian: The Hidden Reality. Parallel Universes and the Deep Laws of the Cosmos(2011).

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This is an old question about the nature of mathematics, is it discovered or is it constructed? Already in antiquity both positions were represented by Plato and Menaechmus respectively, in modern times the debate is in terms of realism versus nominalism about mathematical concepts. The realist explanation is that there is mathematical structure "in" nature, which we discover in the form of mathematical natural laws. Gödel held this view, and it was developed in some detail by Penelope maddy, see her Perception and Mathematical Intuition. She argues that our ability to abstract to mathematical concepts like sets is a combination of an evolutionary adaptation, and developmental learning reflecting historical practice of the community.

The nominalist interpretation is that we construct mathematical theories according to the needs of our practice, albeit filtered through in a diffused way, just as we do our empirical theories. So it is no surprise that the two end up in harmony with each other. The modern nominalist programme advanced especially by Field, is to demonstrate that the use of mathematics in natural sciences requires no assumptions about existence of mathematical objects or properties, rather it serves as a conservative extension of them, that allows for more efficient formulation and derivation of consequences and predictions. See SEP on Mathematical Fictionalism.

There is no reason why both of them can not reflect part of the truth, Dummett gives an insightful discussion in What is Mathematics About?

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