Can mathematics be separated from the physical world?

Question

I am a math enthusiast, with very little interest in physics. In fact, today I thought to myself how can I expel the physical world from mathematics completely. However, this has proved to more difficult than I had previously thought. The notion of a natural number I believe comes from the fact that I can have one or two cups of coffee in the morning. From there I developed the natural numbers system using Peano Axioms (of course with the help of my predecessors), soon after I came to appreciate the real numbers and Euclidean space. My mathematical universe continues to expand on a normal day and it is quite exciting.

However, I have now come to the point where I want to rid myself of the mind restricting physical world, to create a mental universe of mathematics. However, I don't think I can because I started this whole thing by considering my morning coffee. The foundations of the new mathematical system will always be influenced by my presence in the natural world (or at least I suppose this is true).

So, can anyone else see away to deal with the dilemma?

Answer 1

If by "separated" you mean that mathematical objects should be ontologically independent of any single physical object, then I think yes, trivially. As an example, many objects can be descibed as being triangular, but the idea of triangle is independent of any one of those objects actually existing.

On the other hand, it could be argued that if no triangular objects existed, then there would be no concept of triangle... and so the existence of triangle (as an idea) is predicated on the existence of some triangular objects. Presumably, it might depend on the existence of at least one triangular object. This is much harder territory to cover, and possibly the one your inquiry would boil down to. The main trouble in my view with such an approach is this: if there were no triangular objects as far as we can tell, would our lack of such a concept mean that it really does not exist, or simply that we cannot comprehend it? This shifts the focus of the discussion, but you may see where it is going -- questions pertaining to the limits of our knowledge and cognitive abilities. And if we go down that path, we will surely start to inquire about the details of the mind, human psychology and the implication of the modern science of the brain. Further, numerous tangential questions could be asked if one were picky (an example: is any physical body in reality perfectly triangular?).

There have been many discussions about all of this in the past, basically asking whether math is a feature of the world or a creation of man. And there have been many points of view... you might as well start by reading the Wikipedia page on the philosophy of mathematics. In any of the stances described there, it seems to me that it quickly comes to bear that mathematics is something that is thought by people, and communicated between people. Again, any rational discussion will quickly transform into a discussion of not the character of mathematics, but the congition of mathematics, human psychology and, ultimately, neuroscience.

There is another angle to this, a special one in my view, namely the point of view of a physicist. If you mean to ask whether any physical process can be described by several different mathematical objects, then also yes, it usually can and quantum mechanics provides a rich example where various formalisms reach the same conclusions. Even gravitation can be described in different ways... using a purely local formalism, or the more commonly adopted forces at a range, and these will be equivalent in theory.

The same is true the other way around, that is many mathematical objects have proved useful in different branches of physics and science in general. In fact, many mathematical ideas have been adapted to physics after mathematicians developed them without application outside of math. Wigner called this the "unreasonable effectiveness of mathematics", and I recommend reading his and Hamming's papers of the same title. They do not settle any ontological issues for mathematics, but give valuable opinions on the percieved role mathematics plays in science, especially in physics. In particular, we simply do not know why mathematics is so effective in describing the physical world, or why many physical effects can be adequately simplified, and generlized, by such simple mathematical formalae.

Answer 2

This is a very interesting question. I would recommend to take a look at the highest voted questions on this site; there's about half dozen of them asking something like "Are numbers objective" or "Is Mathematics real". The answers to a few of them discuss Platonic versus Intuitionist view of Mathematics.

I'd like to add my 2 cents to that too. Even though it is possible to develop pure mathematics from logic, and it is possible to extend purely abstract notions like that, this approach doesn't lead to interesting mathematics. Mathematical universe is immensely endless, and one can easily follow the path of pure abstraction that leads nowhere.

On the other hand, abstraction that originated from physical world remains very interesting even after the process of abstraction led very far away from the reality. Taking a poetic license, if mathematical though was a ship launched from the continent of the physical world into the endless sea, somehow it would often finds new exotic island far beyond the horizon, long after the continent is no longer visible; if, however, the ship starts in the middle of the sea it almost always wonders aimlessly with nothing notable in sight.

A great example of the above would be Algebraic Geometry. It originated as an extension of Analytic Geometry, a quest to describe the lines and surfaces in 2-D and 3-D given by polynomial equations, something like y^2=x^3-x. This is very well connected to our perception of reality and has tight connections to physical world. Soon mathematicians realized that such equations are easier to study in a projective space over the field of complex numbers (i.e. Bezout's theorem), which pushes Algebraic Geometry away from our perception of reality. Then people realized that replacing complex numbers with arbitrary fields leads to discoveries in Number Theory; then Schemes were developed and Algebraic Geometry departed the furthest shoals of reality. And yet, it keeps produces interesting results that are relevant to the rest of mathematics, such as the proof of Fermat's Last Theorem!

According to V.I. Arnold, "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap."

Richard Feynman was even more direct: "Mathematics is to Physics what masturbation is to sex."

This last quote is very much relevant to your question if understood correctly. Mathematics, taken as a pursuit in pure ideas and logic and abstraction, can be very pleasing mental exercise, just as pleasant as pursuing mathematical physics would be. However, an exercise in pure ideas that were detached from reality from the beginning never produces anything useful. There won't be any children born of purely abstract thoughts.

Answer 3

Yes, mathematics can be separated from the physical world. In Bertrand Russell's words,

So long as it remains pure, it is a game, like solving chess problems; it differs from such games by the fact that it has applications.*

Pure mathematics is a branch of deductive logic.

Deduction tells you what follows from your premises, but does not tell you whether your premises are true.**

Work cited Source:

*Russell, Bertrand. The Art of Philosophizing. New York: Philosophical Library: 1968

**Russell, Bertrand. The Art of Philosophizing. New York: Philosophical Library: 1968

Answer 4

By attributing semantic meaning to emptiness, the foundations of mathematics can be derived without any dependency on a "physical world". This unique approach to iconic representation of thought takes its roots in C.S. Peirce's logical graphs and semeiotic theory. It was refined by G. Spencer Brown's Law of Forms. William Bricken has been the major contributor to this line of thinking in recent times and has even gone so far as to implement the foundations of the alternative paradigm in executable computer code.

A short paper by Bricken, The Mathematics of Boundaries: A Beginning (PDF) describes the construction of both arithmetic and logic out of empty space.

If you want to know if mathematics can be separated from the physical world, I think a good starting point is to evaluate the validity of a mathematics that is based on the absence of any world (physical, mental or synthetically axiomatic). Enjoy.

Answer 5

Since you are living in the physical world, it is just natural that you take your observations from the physical world and use them as inspiration for mathematical ideas. You need to be aware that this works to some degree, but not further.

You think about natural numbers as a mathematical equivalent to "one cup of coffee, two cups of coffee, three cups of coffee" and this is just fine to get some inspiration. But then you want mathematical clarity, and you get the Peano axioms defining the natural numbers. And the natural numbers are separated from the physical world: Given any integer, you can add one, or multiply by two, and get a bigger integer. Count the atoms in the universe, and you can't add one more because you counted them all. The number of atoms in the universe is huge, but not infinite.

In mathematics, you progress from natural numbers to integer numbers to rational numbers to real numbers to complex numbers. And the real numbers don't really work like the physical world, and complex numbers don't work like the physical world at all. But it turns out that physical laws can be expressed in ways that allow calculations and predictions to be done using mathematics. There is an astonishingly strong connection. There isn't any obvious reason why that is so - maybe our intuition coming from the physical world has led us to find only certain kinds of mathematics, and if the physical world were different, our mathematics would be different. Or maybe if the physical world were different, our mathematics would be just plain useless to be used in physics.

When you get to set theory and questions of cardinality, you get into areas that leave physics behind. There is no equivalent to the power set of the real numbers (maybe there is, the set of all imaginable universes), but there is no equivalent to the power set of the power set of the real numbers.

In geometry, Euclid found his axioms that work really nicely with the real world. Except that it turned out that his axioms described three totally different kinds of geometry, of which one matches closely the physical world, one matches vaguely a distorted physical world, and the third is just strange. At that point, I'd say mathematics works fine without the physical world. It can be used to understand the physical world, but it can stand on its own and without it.

Answer 6

Your urge seems to be the otherworldly allure of old-fashioned Platonism, and the very reverse of most modern tendencies, which is nice. In the ancient sense, Geometry indicates a form of completion, purification, or "self-enclosure" that at last eliminates the contingencies and irrationalities of physical experience and presumably transcends its own origins.

This idea, which was last resurrected by Frege and Russell, seems to be in general disfavor now, in part for good ideological and political reasons. By reducing mathematics to set theory and logic, the hope was to make it a self-completing, purely "analytical" system, ultimately a tautological construction grounded in axioms, if not the old faulty Euclidean axioms. (For Russell, one effect was actually to inoculate "reasoning" in the real, natural world from its proclivities towards metaphysical "totalitarian" prescriptions. Analytics can model the world through science, but cannot ultimately "prove" anything necessary about it.)

But if mathematics is to be purely axiomatic, what are the axioms to be founded on? Intuition? Clear and distinct ideas? And how are we to arrive at universal intuitions? The problem is an ancient one that became more acute when it was determined by Gauss, et al. that Euclid's system was not fully deducible from his axioms, and that relatively consistent, perfectly useful systems could be generated from other axioms... though never, it turned out, mutually and completely reduced to one axiomatic system. Of course, Godel is said to have delivered the coup de grace to this hope of completion.

As others have pointed out, physics might take this not as a defeat but as a triumph, for it retains the relative certainty of mathematics, while allowing its uncanny, seemingly unlimited involvement in the physical world. We now have more attempts to derive math from physics, the near inversion of the effort to reduce it to logic. Penelope Maddy, for one, attempts a set theory realism, not unlike your coffee cup process, which might then be correlated to mathematized "neural assemblies."

One interesting "in between" theory would be Kant's once notorious classification of mathematics as "synthetic a priori" as opposed to purely analytical. In other words, it is a priori idealism, not based in physical experience, yet neither is it a tautological system of analytical deductions. Like induction in physics, it produced or "synthesizes" new knowledge... knowledge that is not absolutely "certain" and potentially complete, but is "as certain" as rational processes can get for human consciousness, given its necessary, categorical structure.

There are, of course, many developments of mathematics that are highly speculative, far from any physical origins, and seem to have no physical applications, a possible example being Cantor's infinities or even Godel's platonism. Can the mathematician then follow these increasingly pure forms and wean himself from their physical origins? Cantor went mad, "left his senses," and Godel starved himself to death. So perhaps there is a "way out" for you, if that's what you really want.

Answer 7

I'm not exactly sure which type of math you want to create. If you just want to create a math different from the math suitable for the physical world, then I have an interesting suggestion. Try to imagine a physical world without god. Try to strip away all the reliability and security, all the eternity present in the physical world. Or if you have problems with god, try to imagine a computer not as reliable as the ones we build today, but one which frequently makes mistakes, where the bits flip quite often, or which has some other reliability problems you find interesting. Now try to create a math for such an unreliable physical world or such an unreliable computer.

---

The comments indicate that my understanding of the question wasn't the intended one. Let me try to explain how I understood it: The question sketches how experiences from the physical world naturally lead to the notion of a natural number. In the end the question states "I want to rid myself of the mind restricting physical world", which I interpreted as a reference to the natural numbers, and how you can free your mind of this restrictive concept. I thought there is a point, because the natural numbers are indeed closely linked to the physical world, especially to god as its creator. You can have two really huge natural numbers, and you can tell with absolute certainty whether they are equal or not, even if they are described in two completely different ways. Well, it's actually not you how can do this (answering this question might simply exceed your available computing and memory capacities), but god can do it. (I also have images from Jared Diamond's Collapse in my mind, where he suggests that one ancient culture just dissolved voluntarily, because the environment wasn't stable enough, and they were fed up of rebuilding everything again and again.)

From the comments I get that the goals of the question are actually the other way round. You don't want to weaken the infallible certainty of the natural numbers, but on the contrary extend their infallibility to even more concepts: "then prove to me that you have cut it to the exact size with now error". Perhaps my suggestion to just remove god goes too short. Try to replace it with something better, like a world that tries to fulfill all your wishes, if you just wish hard enough. Or imagine a computer that tries to correct the programmers mistakes in the programs it's executing. Maybe it also can restrain from this to help the programmer with debugging, but if allowed it will be able to understand the intent of the program instead of the actual words in most cases.

Answer 8

You say you went from two cups of coffee to the Peano axioms to the natural numbers. This is, first of all, an extremely strange progression. Most of us became pretty familiar with the natural numbers long before we'd ever heard of the Peano axioms. Fermat, Pascal and Euler managed to discover a great many deep truths about the natural numbers and never learned about the Peano axioms. So if this is really an accurate account of your experience, then your experience is so much at odds with anyone else's that I'm not sure any of us can have much insight into what is and is not possible in your mental universe.

Next, note that just as there's far more to the Peano axioms than there is to the distinction between one and two cups of coffee, there's far more to the natural numbers than there is to the Peano axioms --- as is evident, for example, from the fact that the Peano axioms have (first-order) non-standard models, and so cannot fully describe the properties of the natural numbers. I know (from Frey/Ribet/Wiles) that the natural numbers satisfy Fermat's Last Theorem. I do not know whether Fermat's Last Theorem follows from the Peano axioms.

Because the natural numbers transcend what you can learn from the Peano axioms (and way transcend what you can learn from counting your --- presumably finitely many --- coffee cups) I do not see what would lead you to imagine that they're ultimately based on either of these things, or on anything physical.

It seems clear to me that the primeness of the number 17 is not a physical fact. You could (just barely) argue otherwise, saying that it's a physical fact that there's no way to arrange 17 stones into more than one equally sized pile. But what about the fact that there are infinitely many primes? On what physical observation do you imagine that to rest?

Answer 9

It is necessary to discriminate between that what induces, motivates, or visualizes an idea and the idea itself.

Accordingly I agree with you that physics and even daily experience prompt us to form the idea of a natural number. We formalize this idea, i.e. the concept of natural numbers, by Peano axioms. By this step we have left the realm of physics and entered into the realm of mathematics.

In the realm of mathematics we now make new definitions and prove statements about natural numbers, e.g., propositions about prime numbers. The whole path from natural numbers to integers, rationals, reals, complex numbers, quaterions etc. is a path with in the realm of mathematics. But the motivation do lay out such a path is often triggered by physical needs and insights.

Hence I do not see a dilemma between mathematics and physics. They are two separate domains, each which its own set of rules. The rules of mathematics are independent from the rules of physics.

One has to be always aware whether one stays momentarily in the physical or in the mathematics domain.

Answer 10

I think there are lots of mathematics out there that have a tenuous connection, if any, to the physical world. I'm thinking of abstract algebra or algebras in general. They have correspondences in the natural world, but I don't think they rely upon them in any strong way. Not sure if that helps, but something to look into as you study.

Answer 11

Read the introduction, the preface and the beginning of "The Principles of Mathematics" from Russell. He explains the distinction.

https://archive.org/details/principlesofmath005807mbp

Have a nice day.