It is a rather indisputable fact for physicists that mathematics really is the correct language of physics. Without mathematics one could not formulate physical theories and then make prediction to be tested against nature. Indeed, the formulation of physical theories has required the development of new mathematics. Theoretical physics is really the construction of mathematical models to describe nature.
Even the experimentalist cannot avoid mathematics. One has a lot of analysis of results and statistics to preform in order to make sense of the experiments.
It is rather clear then, that without mathematics one will not go very far in physics. Any understanding of nature is going to be rather superficial without some mathematics.
A little deeper than this I believe that mathematics is more than just a language for physics, or indeed all science. The structures, patterns and rules of mathematics can guide one in constructing/analysing theories. The notion of symmetry is so fundamental in modern theoretical physics and at its heart is group theory. Understanding physics can be driven my mathematical beauty. Given a new theory the first question to ask is what are the symmetries?
One has to ask why mathematics is the language of the physical sciences? Can we understand why mathematics has been just so useful and powerful in structuring our understanding of the Universe?
Eugene Wigner in 1960 wrote an article The Unreasonable Effectiveness of Mathematics in the Natural Sciences which was published in Communications on Pure and Applied Mathematics. Wigner argues that mathematics has guided many advances in the physical sciences and that this suggests some deep link between mathematics and physics far beyond mathematics simply being a language.
A very extreme version of this deep interconnection is Max Tegmark’s mathematical universe hypothesis, which basically states that all mathematics is realised in nature. What this hypothesise also suggests is that the Universe really is mathematical. We uncover this mathematical structure rather than impose it on nature. This would explain Wigner’s “unreasonable effectiveness”.
We are now close to having to think about the philosophy of mathematics and in particular Platonism. I am certainly no big thinker on philosophy and so will postpone discussion about the philosophy of mathematics.
I would not go as far as to say I believe in Tegmark’s hypothesis, but it is for sure an interesting and provocative idea. It certainly makes one think about the relation between mathematics, physics and the nature of our Universe.
Wikipedia Links
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Nice post, ajb.
Tegmark is way out in left field. There is a LOT of mathematics that has nothing to do with physics. There is even more where some parts of mathematics being realized in nature preclude other parts from being realized (the universe cannot have all of the potential toplogies, it can have only one). So to think that all of mathematics is realized in nature is absurd. Eiinstein showed that Euclidean geometry is not realized in nature, even though it is a very useful approximation. There are also perfectly consistent and beautiful theories (Newtonian mechanics for instance) that, while they have been shown to be very good approximations under many circumstances, has ultimately been shown to be wrong.
While I agree that “beauty” is a useful idea in discovering physical law, it is not the final arbiter of validity of physical theory. That role is reserved to experiment.
“It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.” — Richard P. Feynman
On the other hand I have the distinct feeling that a bit more attention to mathemtical rigor would go a long way to improving the state of theoretical physics. QED predicts the anomalous magnetic moment of the electron to something like 14 decimal places — one of the most accurate predictions in physics. But it mispredicts the observed cosmological constant (based on vacuum energy) by a factor of about 10^(120) — one of the largest mispredictions in physics. No one understands this.
DrRocket: I generally agree with what you have said about Tegmark’s idea. However, we may be misinterpreting what exactly what he means and in particular “all mathematics”.
More interesting I think is the idea that the Universe and nature is truly mathematical. One does not impose the human mathematical way of thinking on the Universe, rather we uncover the mathematical structures that are really there. Though my pragmatic point of view is that we may never really make such a clear distinction and philosophising over this is a waste of time.
It should really be no great surprise that mathematics is useful in describing physics.
One of my old math professors once said that “Mathematics is the study of any kind of order that the human mind can recognize.” If one believes that nature is understandable, hence orderly, then mathematics is the natural means of describing it. Saying that the universe is mathematical is synonymous with saying that the universe is understandable.
On the other hand that does not mean that all mathematics is applicable to physics, or that any theory that is mathematically consistent and “beautiful” is necessarily a valid description.
I have readd Wigner’s essay, and have heard him speak in person. I share his feeling that mathematics is successful in the natural sciences, but I do not share his feeling that this success is at all unreasonable. We recognize order and study that order — and that is mathematics.
What does Feynman mean by beautiful? How would a theory be regarded as such if it doesnt correctly mirror reality through experiment?
Appolinaria: “beautiful” is not really easy to describe concretely. Generally I would take it to mean that the physical theory is based on solid mathematical principles, or has a lot of symmetry.
For example I would say that (classical) gauge theory is very beautiful as it is completely determined by mathematics, in particular differential geometry.
Oh okay, thanks ajb.
It’s funny, I just learned what symmetry is in a book I just got by Feynman.
Newtonian mechanics, particularly in the variational form of Lagrangian or Hamiltonian mechanics, is an example of a beautiful theory, because it is mathemtically consistent and so much can be derived from so few assumptions. But Newtonian mechanics has been shown, experimentally, to not correctly mirror reality (particularly when very high speeds are involved).
So is Newtonian mechanics more of a lucky approximation that works in limited situations, as opposed to a perfect set of equations that are unwavering no matter if there is a change in variables?
Sorry, new to ALL of this, trying to understand how it all fits together.
Is it any surprise that as humans we see order in the Universe, when we are the children of an ordered Universe? Any time I have asked these questions and have tried to apply meaning I find my tests focus on an unordered space and I test the continued validity of each hypothesis. This seems to be quite in contradiction to everyone else, who seem to test the validity in a well constructed space. Although I am mathematically limited by my experience, I still find my conclusions lean towards a logical construction throughout the gambit that is both logical and ordered, even when these two qualities seemingly break down. I want to have an opinion, and my intuition tells me that Tegmark isn’t completely left field. I hope to one day be able to make proper claims on this matter in either direction.
Great post AJB!
Appolinaria: Newtonian mechanics works extremely well for bodies that are not too light nor too heavy and bodies not moving at speeds anything like that close to the speed of light. Einstein’s theories are needed for bodies travelling near the speed of light. On scales near the atomic scale or less quantum mechanics is needed.
A little more generally, DrRocket’s point is well made. The Lagrangian and Hamiltonian formulations of mechanics are very beautiful and of great mathematical interest.
Personally, I think max tegmark is on to something. All he is saying is that mathematics is realized in nature and must be uncovered. He, in away, is saying that mathematics have no limitation but it is we who are limited in our understanding. Mathematical structures are uncovered rather than imposed. Just because we do not understand certain structures in nature, or have not uncovered the mathematical ensemble of these structures does not mean they are not present. Just look at the boson, we are coming closer and closer to uncovering their reality, while years ago, we swore they did not exist.
Once again, thanks ajb. That was clear & easy for me to understand.
What Tegmark is saying is that “Reality is nothing but a mathematical structure, literally”. Because only mathematics is the ultimate truth. Dean Rickles in the article (cut and past in browser, I am avoiding spam filter)
http://www.fqxi.org/data/essay-contest-files/Rickles_Rickles_fqxi_2.pdf
argues that if you take away every thing in reality you will be still left with the fact that the Mathematical truths exist. exactly where/how is unkown but such structures could leads to our reality. Hence all structures exist that may or may not lead to a dynamic universe.
But Now comes my theory Quantum Statistical Automata (listed in the speculation subforum) which asserts the hypothesis. But I find that only one dynamic structure is possible and that one comes from a fundamental entity which is a random line (which is really a number).
So, the situation is super bizarre, only one daynamic design is allowed by Mathematics and that one leads to our universe, what a luck. But wait, this only universe had to lead to Us humans what an odd odd.
Appolinaria:
I have no idea whether or not Newton’s equations of mechanics were a matter of luck. I prefer to think that they are a product of inspired genius, but I am not quite old enough to have been there when Newton formulated them.
I will tell you that prior to Newton, while there were fits and starts and deep thinkers (Galileo for instance), there was really no such thing as science. Newton transformed the situation to one in which extraordinarily accurate predictions of the motion of bodies were available based on a very vew fundamental assumptions and the application of the calculus that Newton invented just for that purpose. Newton converted Kepler’s laws of planetary motion from a mysterious set of empirical rules into the first true scientific theory. After Newton, it was seen that nature is understandable, and that was a discovery of monumental importance — modern science was born and became separated from philosophy and magic.
Newton’s equations are not “a perfect set of equations that are unwavering no matter if there is a change in variables” and neither is any other theory that has been invented since. ALL existing physical theories are known to have flaws and to be ony good models of natural behavior under limited conditions. While there has been hope of finding the “ultimate set of laws” for several decades, no such “theory of everything” has been produced, nor is there any indication that it will be produced any time soon, if ever.
There is a lot more that we don’t know than there that we do know.
Yeah, I don’t think I used the right word. I never wanted to attribute Newton’s hard work & genius to luck. With lucky I meant, will future theories likely incorporate aspects of the past ones or completely replace them? Are we ACTUALLY uncovering unwavering reason behind what we see? I just don’t understand why Newtonian mechanics doesn’t work at high speeds.
Don’t mistake me for having interest in discovering the ~*theory of everything*~ However, while I’m here, technically shouldn’t there be something that ties all of these disconnected theories together? Or is that just me having a simple mind?