Monday, December 7, 2020

Difference between Mathematicians and Physicists

For the last 50 years, physics has gotten so mathematical that many people view mathematics and physics as two variants of the same field. The star of this view is Ed Witten, who is widely for both his mathematics and physics.

This is mistaken. Mathematics and Physics are not so similar. Yes, they both use numbers and fancy symbols, but here are three big differences.

Proof v experiment. Mathematics is all about what can be proved from the axioms, like ZFC. The mathematician seeks 100% certain knowledge, and settles for nothing less. The physicists gains validation by doing experiments. Truth is just a tentative shorthand for explaining some observations.

Infinity. All the interesting mathematics uses infinities. The concept is essential to everything. There are no infinities in the natural world. While they occasionally crop up in some physics theories, they are not essential to anything, and there is no reason for a physicist to believe in them.

Spin. Mathematicians are like fermions, and physicists like bosons. Each mathematician is like a unique piece to a giant jigsaw puzzle of knowledge. Physicists do not see things that way at all, as they replicate the work of others and are susceptible to groupthink.

These are huge differences. They are so large that I don't think that it makes sense to say that the fields overlap.

Sure, math gets applied to physics, and there are some mathematical physicists who are really mathematicians in their outlook. But mostly, mathematicians and physicists are different animals.

Dr. Bee writes on whether infinity is real:

Infinity and zero are everywhere in physics. Even in seemingly innocent things like space, or space-time. The moment you write down the mathematics for space, you assume there are no gaps in it. You assume it’s a perfectly smooth continuum, made of infinitely many infinitely small points.

Mathematically, that’s a convenient assumption because it’s easy to work with. And it seems to be working just fine. That’s why most physicists do not worry all that much about it. They just use infinity as a useful mathematical tool.

But maybe using infinity and zero in physics brings in mistakes because these assumptions are not only not scientifically justified, they are not scientifically justifiable. And this may play a role in our understanding of the cosmos or quantum mechanics. This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers.

Infinity and zero are everywhere in mathematics, so if you are applying math to physics, they will be there. But they are only mathematically real, and do not exist in the natural world.


  1. It's rather mathematicians who like to utter zero and infinity in the same breath. But there is a lot of difference of hierarchical levels between these two concepts.

    It's true that both zero and infinity are objectification of mathematical processes.

    But zero is much simpler than infinity. Negative integers (and reals, eventually) are the next higher step, but also quite simple as compared to infinity. Zero makes sense even in the contexts where all numbers directly represent actual physical processes i.e. finite transactions (whether at the account or the system boundaries). That's why, zero can be treated as a number, but infinity cannot be.

    The objectification of infinity becomes indispensable only in the context of calculus---I mean, when it comes the ideas of limits and convergence. This development required a lot of physical observations and reasoning, and then, geometrical reasoning based on them (all being conducted with some valid cognitive purposes to them).

    So, correct me if I am wrong, but I do think that there is no acute (indispensable) need for having infinity in the mere algebra of sequences and series, not even at the stage of stating the general term of a series or a sequence via the general algebraic expression. You can even express a sequence as partial sums of a series, using only the general term, but without using infinity.

    It's only when you come to the convergence of the indefinitely long sequences/series that you can't make do without infinity. The concept of convergence assumes the concept of limits, and the latter requires infinity. Around the same time (i.e. hierarchical level) come the geometrically derived observations concerning convergences.

    Then comes the calculus.

    Most mathematicians turn ultra-sloppy when it comes to explaining the hierarchical nature or roots of their concepts, let alone the motivations for the same or the physical context and purpose in which the observations, motivations, and intermediate conclusions themselves are grounded.

    Now, a word about Indians.

    Many Indians habitually think that zero exists in the natural world.

    Some Indians even think that everything (not just the physical universe but also life and consciousness etc.) came from some "zero". This latter "concept", they insist, is to be taken in the literal sense, as *the* absence of everything, the absence of existence itself.

    A few of them then forge further ahead, and unhesitatingly posit an existence for the idea of Non-Existence, with both Existence and Non-Existence (Sanskrit: "sat" and "asat") being on equal footings, and then insist that *both* these came from the Zero. If you probe them a bit (and sometimes without any such effort on your part), they enthusiastically affirm, in a quite friendly tones, that the Zero acting as the Source of Everything (e.g. all people, animals, mountains, trees, the Sun, the Moon, the lunar mansions, stars, wind, water, everything) was in the nature of a Consciousness.

    On that light-hearted (but conscientiously recorded) note, let me end this note.

    PS: In the Indian tradition, there have been debates with respect to the very last point too, with the other side saying: No, the Zero was in the nature of a *material* thing. Both agree that it was supernatural---or at least, they don't debate on that one point.

  2. Infinity should be jettisoned from physics, to use it in any kind of approximation of a ratio or relation of some kind with something finite leads to utter stupidity.

    Zero is a mathematical abstraction, not a quantity in reality. In reality there is no mystical place holder for the absence of something, you either have something or you don't. In programming it was always considered incredibly sloppy to have output indicating you had '0 quantity' of anything, it was always considered something you should check in your output to be sure it was properly stated as 'none' or 'sold out', 'out of stock', 'empty', etc. You can't have zero dollars, You can be without money or broke. You can't have zero bullets, you can be out of ammunition or have no ammunition.

    Math is not reality.