# In Physics What Role Does Mathematics Play in Determining What There Is?

In Philosophy of Physics: Quantum Theory, Tim Maudlin says the following:

A physical theory should clearly and forthrightly address two fundamental questions: what there is, and what it does. (Introduction, p. XI)

As far as I can tell, describing what it does focuses on the observed effects of matter in motion. Describing what there is focuses on the causes, whether observable or unobservable, that in turn explain the effects.

So one question I have about what seems to be the generally accepted position by at least some, if not many, scientists is, how can mathematical descriptions comprised of abstract mathematical objects representing what it does in a physical theory also be representations of what there is, i.e., actual concrete physical objects, or at least nonphysical objects that somehow are able to morph into physical objects?

I think by using the wave function as an example I can best illustrate where Iâ€™m headed with this question. The wave functionâ€“an abstract mathematical object in a physical theoryâ€“seems also to be equated to some form of an actual concrete physical object, or at least a nonphysical object that somehow morphs into a physical object.

But whatâ€™s the underlying reason for such an idea that prima facie seems odd at best? To my understanding the wave function is used in a physical theory, along with other abstract mathematical objects representing amounts and measurements, to answer questions about what it does.

So what then explains how in a physical theory it can be thought to transition, as it were, from the expression of a quantitative description of what it does in terms of an abstract mathematical object, into an expression of a qualitative description of what there is in terms of a concrete physical object, or at least be able to morph into one?

To me this seems like a rather extreme position that calls for some pretty weighty evidence. Now Iâ€™m not suggesting in the case of unobservables, as the scientific antirealist seems to, that there is no cause, or that we cannot know what it is. And unlike the pragmatist, I donâ€™t think brushing such issues aside as nonessential is satisfactory.

I simply hold to what could be referred to as metaphysical realism, that through abductive reasoning of following the evidence where it leads knowledge to a reasonable degree of certainty can be determined about unobservable physical causes.

But what is the argument for what at least some scientific realists seem to claim that in physical theories abstract mathematical objects somehow equate to concrete physical objects? The only argument Iâ€™m aware of that could possibly be used as support is that the success of the theory regarding what it does in terms of abstract mathematical objects supports the position of equating it to what it is in terms of concrete physical objects.

But on its own that doesnâ€™t seems like sufficient grounds for justifying such an extraordinary position, especially when considering empirical equivalence where there can be two or more theories with equally successful yet unique mathematical formulations that, as far as I can tell, sometimes have some shared elements, yet sometimes none at all.

Setting aside idealism, is there any other support for this position that Iâ€™m not aware of? Or could it be that Iâ€™m getting the wrong impression in the first place about a significant amount of scientists today holding in some way to this seemingly extraordinary position? Or could it be that maybe itâ€™s just not something that most scientists even concern themselves with?

Iâ€™m honestly not sure exactly what it is you are asking. There are correspondences between mathematics and physical reality. Sometimes we understand the math well before we know the physical application (ex: Group Theory). There are some interpretations of quantum mechanics which â€¦ which Iâ€™m better off leaving to a physicist.

This one

I donâ€™t think most scientists care for philosophy and even the more philosophical fields like physics seems to lack this level of interest generally.

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This is the main question. Basically what Iâ€™m asking is about abstract mathematical objects being equated to concrete physical objects. Does that help any?

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Thatâ€™s a good probability.

Whatâ€™s the difference between what there is and what it does? If what it does is behave like a proton, canâ€™t we say that what it is is a proton? What other meaning does â€śprotonâ€ť have?

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Exactly. How something behaves is the only description we have of what it is.

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Getting closer. You might need to distinguish between abstract and quantum. Quantum is still physical, representing mass and energy, but not in the way we expect from macroscopic objects.

Are you suggesting that when it comes to the subatomic world, some particular words might be better as descriptions of what there is than certain mathematical formulas? Why would that be the case?

If not, what is wrong with using formulas to describe what there is? What would we be missing in doing so?

Isnâ€™t what it does about describing how it behaves, and what it is about describing how it exists? A point like subatomic particle would be an attempt to describe a particular aspect of a protonâ€™s existence, would it not? A wave function would be a mathematical description in regards to an aspect of its behavior, would it not?

The way I understand abstract is a part of reality that isnâ€™t physical and has no causal powers. And quantum is physical reality at the subatomic level. Is that what youâ€™re wanting to know?

Certain mathematical formulas describe what subatomic particles do. Why would that also be a description of what they are?

F=ma is a way to describe how an apple behaves when it falls from a tree to the ground, is it not? How would it make sense to use that as a description of what an apple is?

No. I donâ€™t even know what â€śhow it existsâ€ť means.

It does make sense to me, at least in a reductionistic way. Why not? Apples are made of molecules, molecules of atoms, and down to subatomic particles. At some level, all we have is behavior which we can model mathematically. There is no loading up a pallet of quarky and slicing into it to see what it to see what we got. Apart from behavior and math, what would you propose to describe what apples are made of?

I have a book that I wrote in the field of theoretical population genetics. It contains hundreds of equations, and I spent a lot of effort getting the subscripts and coefficients right. Does the post in this thread suggest that it would be helpful to go through and change each equation into a verbal description?

How else would you describe them? Do you think that it is possible to describe subatomic particles in words that convey more than what is contained in the formulaâ€™s? Can you give some hints at what these words might be?

I suppose, but no branch of science gives any answer to how anything exists. All science does is provide models of patterns of behavior.

Since every equation in a book is supposed to halve the number of potential readers, imagine what a record seller your new book would be. 2hundreds times the prior number!

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Iâ€™m not sure â€śwhat it isâ€ť is the same thing as â€śwhat itâ€™s made ofâ€ť. Since everything we physically observe is made of the same few things, then saying we can describe those few things with math doesnâ€™t have much bearing on how to describe an individual macro-scale (like an apple). In other words, a wavefunction tells me nothing about how an apple is different from a pear.

As a few others have mentioned, I think science, especially in terms of mathematics, focuses on the behavior of things. I canâ€™t recall ever seeing mathematics being used as a description of â€śwhat it isâ€ť. A wavefunction is a mathematical description of how an electron behaves and evolves over time and allows use to predict properties of the electron (position, momentum, etc), but the wavefunction isnâ€™t the electron.

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But an apple is something which, when I bite into it, tastes like THIS. And a pear is something, when I bite into it, tastes like THAT. So again, behavior and not some undefined essence.

I agree that there are emergent macro qualities such as chemistry, life, consciousness, or culture, which result in distinctions which cannot be described in terms constrained to fundamental physics or mathematics. I do not find satisfactory, however, the argument that evidence exists concerning physical domains such as relativity, which are available to philosophy, meditation, or acid trips, but somehow not amenable to observation, measurement, or experimentation. Speculation, yes, and that is the fountainhead of inquiry - but speculation is not in itself evidence.

I guess so. I mean you can certainly model the biochemistry of taste, etc. but thatâ€™s not what I would think of if somebody asked me â€śwhat is an apple?â€ť I would talk about it being a fruit, itâ€™s color, what it tastes like, etc. but I seriously doubt I would use any math. Thatâ€™s not to say that each of those properties is completely without mathematical description (like say the phylogenetics, wavelength, and biochemistry, respectively) but that certainly isnâ€™t how anybody I know would answer the question â€śwhat is an apple?â€ť

I guess maybe Iâ€™m missing the point. I took the question to be â€śis an abstract mathematical object representing what something does equated to the actual concrete physical object?â€ť and the answer seems (maybe naively) obviously, no. We use math to describe how things behave, but the model isnâ€™t the thing. F=ma is a description of how matter behaves, but it is neither a description of what matter is nor matter itself.

Iâ€™m just not seeing the connection of what most of the conversation has been about to:

It seems (again maybe naively) relatively obvious to me that mathematical models we use to describe behavior of an object is different than the physical object itself. I assume an actual electron exists and itâ€™s not the same thing as the wavefunction we used to model the electrons behavior. What does that have to do with abductive reasoning about unobservable physical causes?

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