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 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.
    – user935
    Commented Sep 2, 2018 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
    Commented Sep 2, 2018 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
    Commented Sep 2, 2018 at 21:27
  • 1
    It was Hegel’s point that once we recognize a limit we are already past the limit. Such is the nature of Spirit.
    – Gordon
    Commented Oct 19, 2022 at 2:22
  • 1
    Indeed just think about that your brain is just a tiny tiny part of nature, so why it seems your brain can contemplate and further understand the whole entire world containing it? How strange and magic?... Commented Oct 20, 2022 at 5:49

4 Answers 4


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.


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.


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.


The first “light bearer”: man, Prometheus (who has self-consciousness and the ability to reflect) could theoretically bring the whole universe to consciousness and reason, given enough time. (Hegel).

The tragedy is that man the light-bearer has also one foot in the grave (as Freud saw it and for Freud death was final) Freud recognized this.

So history carries reason forward.

  • I am actually not so optimistic about “reason” and man as a creature. But once we reflect upon Spinoza and Hegel it is hard not to get a little excited Forgive me.
    – Gordon
    Commented Oct 19, 2022 at 2:55
  • In 1889, Plekhanov reported, while he and Pavel Aksel’rod were visiting Engels, the latter stated that “old Spinoza was really right in considering thought and extension as two attributes of the same substance” (Plekhanov 1956–58, II, 360) link.springer.com/article/10.1007/s11212-021-09410-9
    – Gordon
    Commented Oct 19, 2022 at 3:36

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