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There is a web of issues pertaining to the theoretical underpinnings of science that I would like to read more about, and so if anyone could take a look at the following claims and questions and recommend me further resources that discuss these kinds of issues, I’d appreciate it. It seems to be a fundamental assumption of science that empirical evidence must, in principle, accord with logical or mathematical theory, and vice versa; that any sufficiently strong observed pattern in reality has a corresponding theoretical description that we can discover and integrate with other theoretical descriptions, and that any sufficiently strong theoretical description can predict patterns in reality we may observe. But why should these things be connected? Note I’m not necessarily disputing that they are connected but asking, how can we explain why they are connected?

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    Maths. Why did it have to be maths.
    – Scott Rowe
    Feb 21 at 20:38
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    That we can discover and integrate any observable pattern is a rather optimistic assessment. Chances are that we cannot do that for most of them. Indeed, we are still unable to "integrate" chemistry with physics, chemical laws so far cannot be reduced to physical ones and many believe that the task is intractable. And we wouldn't know about what we haven't discovered, would we. There is no such "fundamental assumption" in science, there is only methodological attitude. When it works it works and we have a science, and when it doesn't it doesn't. We can only hope that it will eventually.
    – Conifold
    Feb 21 at 20:50
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    Once you have a critical mass of mathematics gleened from experience and hypostatized thru axioms then math can feed off of itself for a long while. But it does have to go back into the real world eventually to refresh itself.
    – Gordon
    Feb 21 at 23:17
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    @ScottRowe: That reference made me smile. "Maths. Very dangerous. You go first."
    – Alexis
    Feb 21 at 23:54
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    Water surveying doesn't work so well.
    – Scott Rowe
    Feb 22 at 2:06

3 Answers 3

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If the question is "how come theory is effective at describing nature", that would be because we make theories to match with what we observe in nature. Whether the observations are first, or the theory is first, then is confirmed by experiments/observations doesn't really matter - we make the theory to align with what see, and we revise the theories as we learn more about nature, to preserve as much as possible of the explanatory power of the theory.

Note, that theories are not necessarily saying what nature is, but only attempt to describe, more or less successfully, what we see. The theories that don't work well are actually dismissed in favor of better ones.

So there is no great magic or surprise here: we make the theories that we found useful to describe what we see.

The reason theory and nature are connectable/connected is that we observe that there seems to be some order, or regularities in nature. That seems required to even try to have theories about nature. So far, we seem to have been successful at observing/explaining some of/exploiting that order. The Universe could have been chaotic without any regularity or order, and we would have been in serious trouble. So at the end of the day, I think what makes it possible to have theories about nature is that there is some order/regularities in nature after all.

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  • “If the question is ‘how come theory is effective at describing nature’, that would be because we make theories to match with what we observe in nature.” But why is it the case that nature and our theories are such that any of them have the property of being able to match what we observe in nature?
    – Joa
    Feb 21 at 20:50
  • @Joa see the later part of my answer: 1. nature has some order ; 2. we make the theories that describe that order. The key may be that there is an order in nature, regularities that we can describe. Otherwise, not sure if we could have theories.
    – Frank
    Feb 21 at 20:52
  • So it's impossible to understand a waterfall because it's chaotic and always changing.
    – Boba Fit
    Feb 21 at 23:51
  • To some extent, yes. The trajectory of each water molecule is probably chaotic enough that we may not be able to predict it perfectly. However, the overall behavior, or the average behavior, probably has some regularity that we can model profitably. As always, we'll do a simplified or idealized model and work that out. But sheer and absolute randomness seems to be precisely what we can't model.
    – Frank
    Feb 22 at 1:04
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The reason is that we live in a Universe that exhibits properties that can be modelled with mathematics. All material objects are made from a common family of fundamental building blocks which interact in a systematic way. If that were not the case- if matter could not be reduced into common building blocks, and if matter and energy and spacetime interacted in ways that varied randomly, then it would be impossible for us to formulate the kinds of theories you find in science.

More importantly, perhaps, if the Universe had been entirely random, then humans would not have evolved, so we would not have been here to investigate it.

As to why the Universe is exactly as it is, we don't yet have an answer to that. So it remains puzzling that so much of mathematics- which seems entirely abstract in nature- seems to reflect the patterns we observe in the Universe around us.

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    It's not really puzzling to me, just because mathematics is quite "ill-defined" (as I like to say) - I mean, "mathematics" is so flexible, we can elaborate "mathematics" for anything and everything we want, in the end. Another point, is that the mathematics doesn't always work so well, actually. Either it blows up with infinities we can't handle, or sometimes a differential equation has multiple solutions, and sometimes one of them is a new particle and sometimes the additional solutions are tossed off-hand. I really am not convinced that maths works so well :-)
    – Frank
    Feb 21 at 20:44
  • @Frank "The unreasonable vexingness of mathematics", eh?
    – Scott Rowe
    Feb 21 at 20:46
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    @ScottRowe - yeah, I'm not impressed by Wigner's paper. You can poke holes in his claims pretty easily, IMHO. Everybody repeats "oh wow! what a miracle that mathematics applies to nature!", but it's just that there is some order in nature after all, and maths is useful when trying to describe a system that has some level of order. No biggie, no miracle.
    – Frank
    Feb 21 at 20:47
  • @frank I think you make a good point. A lot of the mathematical models we develop in physics are based on massive simplifications, or have a limited range of applicability etc etc. You'll find many questions on physics SE which arise because people have treated a particular mathematical treatment as is it were a gospel truth, when in fact it's just a crude but effective model. Feb 21 at 21:18
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    @Frank Well... No. Not a series of arguments. A series of unconnected buzz words. The fact that one thing is round does not mean that we can't describe another thing as flat. Or that the concept of flat has changed.
    – Boba Fit
    Feb 21 at 23:46
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Mathematics works because there are objects in reality and they have properties. When we understand these properties, and express them in mathematical form, we can then predict how they will behave.

We know we have detected such a property when we can predict how things will behave. That is, when a theory is validated.

This does not mean we have achieved absolute and final truth. It means we have observed, at some level of accuracy, that a thing has a property.

A mundane example may help. Consider a bowling ball. We can attempt to predict where it will roll given a certain push to start. One way to do this is to treat it as a uniform sphere. Enter a bunch of mathematics. And guess what? It works fairly well, to some level of accuracy. The ball is spherical to this degree of accuracy.

When we measure things more accurately, maybe it does not work. When we measure the speed and angle of the ball more accurately, maybe we note discrepancy. Look, it wobbles here when spheres shouldn't wobble. So we go look more carefully at the ball. Finger holes! The ball has "non-sphericity." So it's not a perfect sphere. Does the non-sphericity account for the discrepancy? A bunch more mathematics. Yes it does. Again, to this degree of accuracy.

We get more accurate. The ball isn't quite perfectly spherical on one side. More maths. It isn't quite perfectly uniform in its internal material. More maths. Lather, rinse, send back to shipper by return post, repeat. At each step, we measure the behavior as well as we can, predict what will happen based on the properties we suppose, and it works to some degree of accuracy. When it fails we look for new properties.

This process is a big part of what science is about.

Examples abound. Sometimes the nature of the "non-sphericity" is mundane and we are not perplexed by it. The Earth, for example, has hills, valleys, rivers, lakes, oceans, and so on. So we are not surprised that walking distances cannot be treated perfectly as moving on the surface of a sphere. Sometimes the "non-sphericity" is surprising and people resist it. Quantum mechanics still sometimes defeats people who have taken advanced degrees in it. Yet the agreement between experiment and calculation is stupendous.

Calculation and experiment for the magnetic moment of the electron agrees to 10 significant digits. The current limits are on the ability to calculate more accurately, since the calculation becomes very much harder at each order. There is clearly some knowledge of the properties of objects here.

But at each step we are measuring the behavior of existing things. Then we are attempting to make predictions based on hypothesized properties. If the predictions are successful, we have detected a property, at least to that degree of accuracy.

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  • The requirement for order or regularity would need to be included though. It's a required premise to have any kind of law or theory. If a property does not have any kind of regularity about it, if it is purely random in a strong sense, then we can't make a theory for it. We may make a theory for the average, if the average follows some law, but having a theory seems to require some order/regularity.
    – Frank
    Feb 21 at 22:39
  • @Frank And then there are theories that the properties change over time. For example, we are currently playing with cosmology in which the speed of light is changeable. "Property" includes within it the notion of the degree to which it changes over time. Order and regularity would be properties within certain theories.
    – Boba Fit
    Feb 21 at 23:42
  • I think the point is rather than "random/no regularity/no order" is precisely equivalent to "cannot be modeled" assuming model stands for theory here. If the property you have in front of you is truly random, you will not be able to model it in any way (because any model would mean it's not really random).
    – Frank
    Feb 22 at 1:14
  • "Mathematics works" is a hand waving generality ;-) "because there are objects in reality" is a gratuitous assertion, that has been the subject of volumes of philosophy. For "mathematics works", see my discussion with Marco, which shows that, among other things, physicists can choose what to include in their models partly as a function of the maths they have at hand. You say as much yourself, when you give the example of an iterative refinement process, implying a succession of approximations.
    – Frank
    Feb 22 at 1:33
  • Also, by the way "Mathematics works because there are objects in reality and they have properties." just moves the question from "why does maths work to describe nature" ("works" would need to be defined), to "why does maths work to describe properties", IMHO.
    – Frank
    Feb 22 at 2:22

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