Mathematics vs Physics



Posted: Wednesday, September 22, 2010

by Johan Hansson
FunPhysics

Mathematics isn't science. In fact, its approach is exactly the opposite compared to, for example, physics. Mathematics is an elaborate game, in which it is decided from the outset to play by the given rules. Then one proceeds by proving theorems; truths already implicitly hidden in these rules. In that way, one might say that mathematics is the easiest thing in the world, as everything is given from the start - but of course it requires incredibly sharp brains to lure out and understand what is already embodied in the rules - the starting assumptions ("postulates") that neither can be proved nor disproved .

In physics, the situation is just the opposite. We are observing the world and trying to understand how it works, that is, trying to find the rules that nature follows. At least we assume that nature follows rules. But the strategy is difficult because we never have perfect knowledge about anything. It's a bit like covering large parts of a chess board, and then trying to get someone (ignorant of the subtleties of chess) to figure out the rules, by only getting access to some of the 64 squares on the chessboard. For instance, say that you can see only three of the squares; it will be almost impossible to obtain the complete set of chess-rules. No matter how long you stare at the three squares (during a real game of chess on all 64) you will hardly be able to draw any conclusions about what the pieces are doing and why! - when you aren't seeing them. They show up seemingly at random, and leave your three squares just as erratically. We could hardly gain knowledge of the complete rules of the game.

But it works roughly the same with the fundamental laws of nature, with the difference that the number of "squares" we need to observe probably is considerably larger than 64. While some physicists believe that we soon will be able to see the equivalent of all "squares", others think that we still see only three or even fewer.

What, then, is the similarity between mathematics and physics?

A large part of modern mathematics was developed by Newton in the 1600s because he needed it to analyze his laws of motion - mechanics. More modern mathematics, such as group theory and differential geometry, however, was developed "for its own sake", but was soon borrowed and used in quantum physics and general relativity.

This led the renowned physicist Eugene Wigner, in 1960, to write an article entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Why, wondered Wigner, is mathematics - often designed only for its own sake (fun!) - so successful in describing phenomena of nature? Perhaps nature itself is mathematical in essence?

I think it is much simpler than that. Mathematics is a language developed by the human brain. Albeit the most precise language we know of today, still basically a language. Mathematics is a kind of map over how our brain works, with logic as its compass. In the future it may well be that we find (invent) an even more precise language than mathematics, which will then be used to describe all sorts of phenomena. Since both mathematics and science are the fruits of the human brain, it is also perfectly natural that they should fit like a glove. To say that mathematics is "unreasonably effective" for use in physics and other sciences is rather like saying that our tools to make gloves are "unreasonably successful" to make gloves. The book of nature need not necessarily be written in mathematics any more than a computer manual need to be written in English.

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www.FunPhysics.org

FunPhysics.org is a resource for everything on physics, astronomy and the universe, featuring physics researcher, Prof. Johan Hansson - among many other things the discoverer of Preon Stars, as covered in e.g. Nature and Physical Review Focus

Prof. Johan Hansson is a researcher and PhD in theoretical physics, active both in the USA and Europe. He is, among many other things, the discoverer of Preon Stars, as covered in e.g. Nature and Physical Review Focus.

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Top-level comments on this article: (2 total)
» left by Ben Morrish
 1 year 237 days ago.
48 fans.
"They show up seemingly at random, and leave your three squares just as erratically. We could hardly gain knowledge of the complete rules of the game. But it works roughly the same with the fundamental laws of nature" ... except that in the "squares" we observe in the real world things tend not to show up randomly or erratically - we've noticed things show up with reliable predictability, and from that we have been able to gain a better understanding of "the rules of the game". Even if we don't yet know the exact rules of the game, we know that our models of the rules are increasingly close to them. Even in the chess example you give, where we could only observe 3 squares of the board - we could not work out the full rules of chess, but what we observed would not be entirely random or erratic. We could see enough "order" for us to suspect there was a system of rules underlying what we observed, and enough for us to start work on determining what those rules were. Over time, we might find we could observe more squares (just as over time we can now observe more of the real universe) and we could then refine and improve our models of what the rules are. Mathematics is the only language with which we can describe the workings of the universe with the incredible accuracy that we demonstrably can today. There is still more refinement to be done of course, but we are making sure progress - and that would not be possible if our observations of reality had all been random and erratic. Refinements of mathematics would still BE mathematics. Unlike spoken human languages, the language of mathematics is universal.
» left by andy mayor from united kingdom 1 year 221 days ago.
You seem to omit the fact that mathematics is a generic name and it has two sub fields. Pure mathematics and applied mathematics. It looks like you seem to refer pure mathematics in this article because applied mathematics can be as practical as physics is. Applied mathematics studies the same (and even other science subjects) physical phenomena that physics does. This is the subfield of applied maths called mathematical physics. Further applied mathematics has so many real world application beyond physics e.g. bio-mathematics, financial mathematics, operation research, actuarial science, industrial mathematics etc. These are all practicle subfields of mathematics. You should have used the title pure mathematics vs physics instead!
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