## The Role of Mathematics in Science (/Physics)

Discussions on the philosophical foundations, assumptions, and implications of science, including the natural sciences.

### The Role of Mathematics in Science (/Physics)

It is possible to have absolute knowledge with regards to the truth of mathematical propositions concerning abstract entities. Knowledge if the physical world however, may only be obtained empirically. This means there is always a degree of uncertainty with regards to our knowledge of the physical world, for though an experiment relentlessly turns out one way, why should we not suppose that perhaps another day it will turn out the other?

But is it fair to say that one could feasibly obtain absolute knowledge about the physical world, by associating mathematical abstractions about which we can determine propositions of unassailable truth, with features of the physical world? Achieving a perfect description of physical reality is then just a question of finding better and better abstractions, until the abstractions themselves recreate physical reality. Whilst this may not be possible with regards to mathematical abstractions in the sense of contemporary and historical mathematics, computer code is deterministic and in this sense, is not a computer simulation also a mathematical theorem? And is there not more hope for computer simulation in achieving this task of recreating reality than conventional mathematics? Or might it be necessary to find a combined approach, with mathematical abstractions describing entities like space and time, and computer code simulating the actions of matter under such a mathematical framework?

How does / might mathematics and computer simulation describe non-deterministic (e.g. quantum mechanical) features of the universe?

kStro
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Certainly math, as a language, is self-contained and allows for a degree of concreteness. In fact, as far as I know, math is the only branch of science for which there is an unambiguous definition of "proof". Computer simulations and models are also, as you say, deterministic and ultimately simply extensions of pure math. Consequently, since these things are deterministic, pure deduction, etc., applies.

Where this all starts to weaken is when there are attempts to link these things with "the real world". It may well be that, for example, one ion of sodium plus one ion of chloride equals to ions still or, alternately, one molecule of salt. The problem is that, for quantification to work and tell us things, we need to start being careful about what is being quantified (and then manipulated mathematically, etc.), how it is being quantified, etc. Mathematical and computer simulations and models are simplifications of the "real world" so that we can say some things about that real world. We can and do try to make these models more accurate by adding in more factors, more data, more algorithms, etc., but ultimately it is not the computer or the equations, etc., that decides what goes into all of this, it is the individual researcher, relying on judgement, etc., who decides what counts as appropriate date, how it fits in, decides on the appropriate formulas, etc. And then that researcher (hopefully) tests that data against the real world observations, designs tests which ideally employ independent factors which in turn may be different simplifications of the real world, etc. Math may well be a rigorous application of a specific language and structure but it is individuals who decide what numbers and formulas to use etc.

Forest_Dump
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Forest -

Ions are fairly complicated systems. But in linking mathematical abstractions with much simpler, fundamental constituents of the universe, might one not hope that a goal of perfect knowledge might be realized? Once one has perfect knowledge of the fundamental constituents of reality, one can simulate up to describe more complicated things, like ions and molecules and cells and animals and people.

I suppose it depends on whether one has confidence that the fundamental constituents of nature are "simple" or behave in a way that allows us to make the required isomorphisms between mathematics and physics. The quantum regime seems to behave fairly horrifically; I don't know how much confidence that affords one in the realization of such a program. But maybe one can be more hopeful if one considers that advances in mathematics are perquisite to advances in physics, and maybe the mathematical developments that are required to make sense of quantum physics just haven't been revealed yet.

Of course, there's also a requirement in believing that there exist fundamental constituents to physical reality; that at some eventual level you can't physically scale down any further.

kStro
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Quantum mechanics can be formalised within 20th century mathematics. The development of quantum computers may make it possible to simulate large quantum systems.
Whether quantum mechanics is the end all and be all of the universe is a question for physics, not mathematics.
Phalcon
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I don't pretend to know anything but I believe some (e.g. Penrose) still think quantum physics needs fundamental modification that it may sit better with relativity. Quantum gravity isn't really figured out yet I don't think.

kStro
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It is true that currently there is no good theory of quantum gravity. It will require new physics and new mathematics to create one. However there should be no need to rewrite the foundations of mathematics.
Phalcon
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### Re: The Role of Mathematics in Science (/Physics)

kStro wrote:It is possible to have absolute knowledge with regards to the truth of mathematical propositions concerning abstract entities.

The short and simple answer is that by definition metaphysics can never be proven so the answer is yes. It is possible, but whether or not it will ever be achieved or proven is another story altogether.

It reminds me of Ursula LeGuin's short story, "The Lathe of Heaven". In it the dreams of the main character, George, keep changing reality and nobody but his psychologist knows this happening. At one point George says that for all he knows there are other people who can do the same thing, and reality is constantly being pulled out from under all of us all the time.

Does the universe obey mathematics, or do we change it anytime it disobeys mathematics? Is quantum mechanics a description of the "real" world, or of our collective unconscious? Such questions are beyond anything but statistical truth and, as the wit once said, "There are lies, damn lies, and then statistics."

wuliheron
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I posted an argument along similar lines here: http://www.philosophychatforum.com/bulletin/viewtopic.php?t=12014

linford86
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### Re: The Role of Mathematics in Science (/Physics)

kStro wrote:Whilst this may not be possible with regards to mathematical abstractions in the sense of contemporary and historical mathematics, computer code is deterministic and in this sense, is not a computer simulation also a mathematical theorem?

I have to say, this is actually a pretty profound insight. In general, it isn't clear what the relationship of computer code to mathematical theorems is. There, is, however, a beautiful (and rather abstract) special case. Check out the Curry-Howard Isomorphism. And if you are like me and found this totally awesome then you might try learning haskell.

kStro wrote:And is there not more hope for computer simulation in achieving this task of recreating reality than conventional mathematics? Or might it be necessary to find a combined approach, with mathematical abstractions describing entities like space and time, and computer code simulating the actions of matter under such a mathematical framework?

I have a friend who is a graduate student in physical chemistry, and I'll tell you what she tells me regarding this. She says that it is now rather typical to hear a talk in physical chemistry where the authors begin by deriving foundational results in quantum mechanics, and then eventually turn to computer simulations to supplement their data. I understand that this is also typical in evolutionary dynamics (in particular, Martin Nowak of Harvard is a master of finding closed form solutions when possible, and then turning to computer simulations when a closed form solution is not forthcoming). I have seen examples of authors in theoretical economics also employing this approach.

xcthulhu
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### Re:

Forest_Dump wrote:In fact, as far as I know, math is the only branch of science for which there is an unambiguous definition of "proof".

While I agree that the standards for proof in mathematics are largely homogenous... it might not be as completely homogenous as you think. This summer I will be involved in a research project known as Flyspeck. The idea is to produce a completely formal proof, checked by a computer for correctness, of the Kepler Conjecture. The Kepler conjecture states the the densest sphere packing in 3-space is $\frac{\pi}{\sqrt{18}}$. It involves a computer checking that several thousand linear programming problems involving 100 variables apiece are infeasible, among a dizzying number of other daunting results. The author/project-coordinator, Thomas Hales, expects that the whole project will take 20 man-years to complete (that is, 20 dedicated researchers 1 years, or 1 dedicated researcher 20 years). He feels that roughly 10 man-years have been dedicated to the project already, and that the whole project is roughly 50% complete.

Not all philosophers of mathematics agree that, when completed, Flyspeck will really give a proof. For instance, Reuben Hersh writes in the beginning of his anthology 18 Unconventional Essays on the Philosophy of Mathematics that
Reuben Hersh wrote:I do not know anyone who thinks either that this project can be completed, or that even if claimed to be complete it would be universally accepted as a convincing proof of Kepler’s conjecture.

xcthulhu
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