The Unreasonable Effectiveness Of Mathematics In The Natural World
All of our physical theories are based upon mathematical objects (ranging from differential equations to tensors) which initially appear to be human fabrications, but the more you study the physical world, the more it becomes difficult to distinguish between the physical and abstract mathematical entities which embody a theory.
Eugene Wigner summed this observation up as “The unreasonable effectiveness of mathematics in the natural sciences”.
http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
As Wigner discussed in his article, it is tantamount to miracle, that a human construction can so easily and deftly describe the natural world. The implication from this being, that the premise that mathematics is a mere human construct is flawed. I emphatically agree with the idea that there is more to mathematics than meets the eye.
Consider General Relativity. No doubt just about everyone would associate this theory with Einstein, but Einstein’s contribution was not the sole one, and I would argue, not even the most important. Consider the mathematician Bernhard Riemann who lived about 50 years prior to the development of General Relativity. Riemann developed a branch of mathematics called differential geometry which deals with the geometry of entities called differential manifolds in multiple dimensions. Differential Geometry was fundamental in constructing the relativistic model of space-time. The really interesting fact is that Pure Mathematicians, not physicists were the first ones to “discover” the techniques which would allow Einstein and other Physicists to create general relativity. The only way to understand general relativity is to understand the mathematics, and yet Einstein contributed very little to the mathematics of his own physical theory.
On the other hand, Mathematicians like Newton and Maxwell discovered/invented mathematics in their search for physical understanding, but I believe this supports my initial point, as evidently the search for physical understanding has led to mathematics.
It thus seems no mean exaggeration to claim that physics cannot be done without mathematics. Yet this could not be further from the perspective of a layman. To a layman a physicist uses equations to understand the workings of entities like black holes or Quarks, whilst a mathematician plays around with numbers or spends his/her time solving hideously long equations. In the not so distant past, if you had asked a physicist whether he/she was a mathematician or not, they would have returned a look of puzzlement, how could you have taken them for anything else?
Many, including ironically a physicist named Max Tegmark have postulated that our Universe is inherently mathematical, that to understand anything one must be a mathematician, and in many respects I am inclined to agree. The only alternative is that our minds have evolutionarily developed in a process to understand a non linear world, to function in a linear manner, mathematics being a product of this. If the latter is true, than It has no different consequences to that of a mathematical universe, at least for us, as to understand the universe we would still have to use our minds and hence mathematics.
I am interested to hear Jason’s (and anyone elses) perspective on this.
Paul Rosenthal
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Here is Tegmark’s argument for a mathematical universe if anyone is interested:
http://arxiv.org/pdf/0704.0646v2.pdf
A note: Tegmark does not just believe the universe is inherently mathematical, but that it is mathematics.
Paul Rosenthal
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I tend to agree, I think it would be a very odd thing if our mathematics are able to describe our universe and make predictions so accurately if it did not somehow reflect something real within the universe.
Heather B.
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I personally have an issue saying that something is an inherent part of anything (or vice versa), but that’s just me. I still only view maths as a system (or as a tool as my matric physics teacher would say) that allows us to put things in a context that we can understand. And the idea that the universe is maths… I need to think about that one more…
This is an interesting topic. I can’t tell you exactly what I think about the topic as a whole because I don’t know what I think, despite having tried to decide what I think for the past several decades. We should discuss it in tutorials.
I do think that the claim that the universe IS mathematics makes no sense if taken literally. Mathematics is a body of social activity, and the universe is not a body of social activity! But there may be very very close links between the two. That’s where I get stuck.
Jason
Evidence the universe is a simulation.
Jeff A
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For some reason, Jeff’s statement made me think of Herman Hesse’s “Siddhartha” and the idea that the world is just an illusion…
Anyway, here’s an article I found that explores that the idea that we are living in a visual reality, from an information processing standpoint. A very interesting read.
http://arxiv.org/pdf/0801.0337v2.pdf
Sally
Hi Sally – Herman Hesse’s Siddhartha is a wonderful wonderful tale! If you haven’t read ‘the glass bead game’ or ‘Steppenwolf’ by Hesse, they are also must reads
Julien
Also on ‘Wigner found mathematical effectiveness in the natural science to be unreasonable,’ (paul)
Wigner held mathematics effectiveness was not merely a coincidence and therefore must reflect some deeper truth about mathematics.
Yet I don’t see why he found its effectiveness so unreasonable.
If the mathematics used to describe nature is a carefully coded expression of our experience, then its effectiveness is built into its design, and it is perfectly reasonable.
Mathematics is unavoidably anthropocentric, which makes perfect unabashed sense, as maths rests ultimately on the human experience of nature.
Given this, why should it be miraculous that mathematics, a human construction, can so easily describe the natural world?
Consider group theory, the logic of symmetry:
There is a heap of further info into how math encodes our experiences of identification; if loy look into group theory..
in short:
Group-theoretic symmetry is the mathematics used to describe nature, it fundamentally expresses our experiences, codifies these experiences as the axioms of scientific inquiry.
As such mathematical symmetries capacity to accurately describe reality becomes comprehensible rather than mysterious.
Julien
Concerning Tegmark’s ‘Universe is mathematics’:
Tegmark’s claim that ‘physical reality is a mathematical structure’. That is, that the universe is mathematics’: seems to me platonistic in the extreme, positing not only the existence of mathematical entities, but denying that anything exists EXCEPT mathematical entities.
which i think is silly. And i believe that as Jason said, mathematics is at least in part a human construction, as a body of social activity…
The mathematical objects usually posited by Platonists are, unlike concrete objects, supposedly entirely divorced from human experience; we cannot perceive them, nor do they bring about any perceptible effects. I don’t know what nature of mathematical objects Tegmark posits.. but if they are abstract, it isn’t easy to see how we might be capable of knowing anything about these mathematical objects….
It seems more convincing to me that our knowledge of mathematical objects is best explained if these objects are actually not discovered, but created by us, as products of human mental activity.
i’m still very unsure of my position on mathematical realism though, So shall say no more silly things before I think about it a bit more.
(would be happy to hear different arguments on mathematical realism/anti realism in general and Tegmar’s extreme brand of Platonism in particular).
Julien
— Mathematics most certainly is not a human construct. I agree, mathematics is a product of our brains experience with the natural world, but it also takes the interaction to a different level. The axioms of mathematics are created by Humans, I agree, but mathematics is not the study of these axioms it is the process of drawing necessary a priori truths from these axioms. A mathematician is constrained just as any other scientist by a framework which is beyond social convention. Once you define what a group is you cannot say Lagrange’s theorem is wrong, in any sense at all, you cannot argue that it is subjective, because by definition the truth follows from the axioms.
The best argument for platonism is one I read by Hilary Putnam. If you accept physics as “real”, then you must accept mathematics as “real” too. Our theories of physics are mathematical statements (Most are differential equations), and if you claim that our physical laws are only approximations to some better truth, eg. the metric tensor of Einstein’s relativity is a human construction to something beyond any other form of explanation, then they are not reality.
People have a vast problem with the mathematicalisation (Is this even a word?) of science, which I think stems from either an experience with terrible math educators (there are quite a few) or some deeper social trend which dates from Aristotle’s methods of science.
-Paul Rosenthal
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