What proportion of Math is grounded in Physics?

Question

This came up as a tangential in relation to whether math is a thing we're discovering, or inventing:

Suppose we meet with an alien race: their symbols, and even ways of computation might be different than ours, but putting 2 pebbles, and 1 pebble on a table will map to their concept of "3". This is because physics provides a grounding for experimentation on various theories. On the other hand, there is a lot of "fanfic" in Math -eg category theory's Monads- for which there is no such easy grounding. On the middle hand, there are things which can be mapped to the 1st category, and therefore have grounding by relation.

This question pivots mostly around the term "grounding", so for the purpose of this question, I'd define grounding as: results being convertible into physical observations (which do not, by themselves contain the computation itself -no computers re-implementing the same thing allowed!).

Using this (or if ill-defined, closest by intent) definition, what proportion of Math is grounded in Physics? To put it in the larger context, what is the lower estimate on intersection of potential common mathematical understanding between 2 alien races?

Edit: okay, this thread seem to have a problem of gravity wells. I'd like to highlight, that this is specifically not a "bring your pet theory to work" topic.

For this reason, I've added a bounty.

Bounty, and answer acceptance specifically goes for an answer, which has the following structure:

• describes a reasonable methodology (of fermi estimation, of approximation methods, of any reasonable form of method), and a list of specific assumptions it makes.
  • For example: you can ask physicists of all the math they've learned what proportion do they use during their work; deduce what proportion of "all math" is being taught to physicists, and thereby reach an estimate. This is totally a thing you can do.
• works through the methodology from the specific assumptions given, to reach an estimate conclusion
• concludes with a specific number (this was the question), optionally with an upper, and lower bound.

Numerical, quantitative methods, inspired by sociology, economics, or math very much preferred; so is grounding things into numbers as much as possible; methods, or possible ways of attack will also be upvoted. Please kindly keep a firm quantitative perspective when posting in this topic. Thank you.

Answer 1

I don't think a quantitative approach really works here.

An important question for me, given how prevalent mathematics is in contemporary physics, is how much of physics can be motivated without appealing to mathematics.

The milisian monists and the greek atomists developed the shape of physics within broad contours without appealing to mathematics: the unity of the world, its lawlike nature and its divisibility into atoms.

This still holds in the contemporary situation, for example in quantum mechanics there is the notion of the quanta which is a discrete or atomic unit; in string theory even space-time is considered as atomic; if space was 2d, then one would have a picture similar to a carpet with tufted threads.

The role that mathematics plays in physics is complex, and to be honest a quantitative approach yields little understanding: Classical Mechanics was discovered alongside calculus; symmetry considerations (amongst others) prompted Einsteins discovery of GR; matrices, a pre-existing mathematical formalism was used by Heisenbergs in his version of QM.

More recently there has been a convergence of interests in mathematics and physics, in the 70s it was discovered principal bundles gave a geometric interpretation of Yang-Mills theories - hence of EM, QED & QCD; and in String Theory there has been a huge cross-over even with something as esoteric as arithmetic geometry.

Answer 2

Without going into philosophical aspects you could say that almost all mathematics has evolved due to the need to solve some physical problem. Common mathematical understanding would be based on whether or not the civilizations are facing common types of problems .

For instance calculus evolved because people were interested in learning the nature of slopes of curves . But for a civilization of ants this information would be meaningless because they don't have any understanding functions or curves . Their mathematics would be limited to primary level counting.

In fact you can observe this in daily life . There are 2 types of calculators - Normal and Scientific but most average people may not understand the utility of all the functions in the scientific calculator . Does that imply that Human mathematical knowledge is limited to only the operations which can be performed but normal calculators ? Certainly not .

Therefore it is difficult to comment on the intersection of common knowledge of different species. Problem solving ability develops when a problem is actually encountered .

Answer 3

None. If that alien race has only the sense of smell, they will not tell one pebble from another if the pebbles all smell the same.

Take, for instance, numbers: when you count, you count "things," but "things" have been invented by human beings for their own convenience. This is not obvious on the earth's surface because, owing to the low temperature, there is a certain degree of apparent stability. But it would be obvious if one could live on the sun where there is nothing but perpetually changing whirlwinds of gas. If you lived on the sun, you would never have formed the idea of "things," and you would never have thought of counting because there would be nothing to count.

--Russell, Bertrand. "Beliefs: Discarded and Retained." Portraits From Memory. New York: Simon and Schuster, 1956. 40. Print

Mathematical models to the physical world are topographic maps to the actual terrain: the former are just approximate representations of the latter. As precision increases, the smoothness of the representation disappears. Large-scale maps reveal rough terrain; hight-precision measurements result in jagged curves. Take Ohm's law for example: anyone who studied electronics long enough will realize that Ohm's law is not a law at all; it is a rough description of a very special kind of conductor. High-precision measurements reveal that most conductors are better described by I-V curves.

The question that naturally follows is this: how much of today's particle theory is based on mathematical models whose own premises remain rough and unexamined? A physicist should always be a geographer first and be vividly aware of the chains of inferences and the assumptions behind a mathematical model. Probably a generation of physicists brought up in the Russellian school is needed in order to sort things out.

As regards points, instants, and particles, I was awakened from my 'dogmatic slumbers' by Whitehead. Whitehead invented a method of constructing points, instants, and particles as sets of events, each of finite extent. This made it possible to use Occam's razor in physics in the same sort of way in which we had used it in arithmetic. I was delighted with this fresh application of the methods of mathematical logic. It seemed to suggest that all the smoothness of the concepts used in theological physics could be attributed to the ingenuity of mathematicians rather than to the nature of the world. It seemed, also, to open an entirely new vista on the problems of perception.

Russell, Bertrand. "The External World." My Philosophical Development.New York: Simon and Schuster, 1959. 103. Print.

Answer 4

From an intuitionist point of view, I would argue that all of math is grounded in physics via genetics. We have the math we have because we have the intuitive assumptions we share. We evolved those assumptions to fit our environment. Our environment has a physics. So our assumptions are an adapted approximation of the right kind of thinking in which to frame basic (Aristotelian-style) physics.

But the parts of our intuition refine one another when we bring them together in mathematics. We can put together measure and pure geometry and get to the analytic form of conic sections, for instance. And without that combination it would have been hard for Newton to notice that linear motion, ballistics and orbits can be captured by the same set of rules.

From this point of view, mathematics is the ongoing exploration of how those intuitions do or don't cleanly fit together, and why. So farther from physics, math is still based on combinations of intuitions that apply to reality. We have monads because we have groups, we have groups because we have fields, we have fields because we have Galois' original observation on the permutations of roots, we have those because of a specific obsession with polynomials, starting from the analytic forms of conic sections, which became especially important when Newton unified gravity with the inverse square law.

I would think that any two species that occupy the same shape of space would be well served by very similar Aristotelian-style physics, and would evolve quite similar patterns of thought to maintain those. But a race that moves, for instance at speeds we would consider very high, or who navigates a mostly solid environment rather than a fluid one, might have a very different kind of basic mathematics, because so much of ours is based in intuitions about time and empty space.

Actually, the 'fanfic' may be more similar. Algebra as a whole, and arithmetic as its special case, capture patterns that ordinary things fall into by combination, permutation or symmetry, and any model of space and objects probably still has those features, even if it lacks any actual instances in common.

I would bet that at a high level, we and they would share something as esoteric as the classification of finite simple groups, for example, although for totally different reasons -- we got there from roots of polynomials, but then we applied the concepts all over the place. They might get there from Polya groups of chemical crystals, or the knot groups of polymer strands, or something, if they are a non-Euclidean race.