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In studying math we are, at least partially, interested in making abstraction of real world problems and solving them through rigorous techniques and methods, and then interpreting the result. Let us zoom in to one such problem - Suppose we were to create some kind of a model (perhaps a dynamical system of some sort) to govern a phenomenon we observe in real life. In constructing this model, we create equations to represent quantities and relationships in the system that are of interest to us. Once the model is in place, we go about 'solving' it for relevant quantities and then interpreting from there.

My issue lies in the detachment of the physicality of the problem during the process of solving the problem. When we solve such systems we often use techniques such as complex integration, or abstract matrix or operator manipulation which, as far as I understand, is a necessary level of abstraction that we apply in order to obtain solutions. Its as if we start the problem in the physical realm where the equation has a direct correspondence with the problem, then move to the mathematical realm where we do manipulations. In this moment we (or maybe just I?) lose sight of the actual problem and work only on the math of it, and then we convert back. What happens in this middle area and how am I to rest assured that these abstract manipulations aren't fundamentally changing things or introducing new nuances that I need to care about?

Yes I understand that the math is all quite formal, and we 'trust' it, but its almost feels like a question of scope. While working with the equations I personally often loose sight of the problem until I'm near a solution.

Also, it seems easy to look back and say, ok well our solution to some model gives valid results every time, so of course we can trust this mathematical process, but surely there's a more sound explanation. Perhaps its my personal nearsightedness that's interfering in the process, but regardless I'd like to hear some other peoples thoughts on this.

PS : I recognize that this might be borderline off topic, but its bugged me forever and this seems as good a place as any to ask, please feel free to redirect.

Thanks!

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closed as off-topic by AlexR, Thomas, Niels Diepeveen, Nicholas R. Peterson, TZakrevskiy Feb 23 at 18:39

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This seems to have more to do with philosophy of science than math. In science we like to build models that fit the data. If the model doesn't give valid results, we throw it away and find another one. –  Tyler Feb 23 at 16:54
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What mathematics is, whether or not it can be "trusted", if it is universal etc. are ancient and deep philosophical questions without a clear answer. –  naslundx Feb 23 at 17:02

3 Answers 3

Its as if we start the problem in the physical realm where the equation has a direct correspondence with the problem, then move to the mathematical realm where we do manipulations. ... What happens in this middle area and how am I to rest assured that these abstract manipulations aren't fundamentally changing things or introducing new nuances that I need to care about?

In my opinion, you cannot rest assured. You can simply say "it has worked well so far". Consider the example of the simplest mathematical model: using numbers to represent collections of objects. There is always a possibility that tomorrow, in the physical universe, one apple and one apple might make three apples, and though physicists might be surprised to find that energy is not conserved, we don't have grounds to complain "but that's not how it works in mathematics".

Consider this quotation, which of course is more generally applicable than the issue at hand:

"The map is not the territory." - Alfred Korzybski

For us, the map is the mathematical model, and the territory is the physical universe. We can try to draw an accurate map, and then (for example) try to draw a path between two parts of the territory, but if we try it out in real life and it turns out to be wrong, we are not surprised by this, nor do we complain that making marks on a piece of paper should have logically implied that a statement about physical reality should be true. We say "our map was inaccurate".

Here are some more relevant quotations:

  • "The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success." - Paul Dirac

  • "All models are wrong, but some are useful." - George Box

  • "There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain." - Alfred North Whitehead

  • "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." - Albert Einstein

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This is a bit abstract but if you for design your model in a way that a mathematical property $A$ (for instance a formula or whatever) holds and is by crazy weired stuff (usage of fix point theorems or whatever) possible to show that $A\Leftrightarrow B$ while $B$ is especially easy to interprete. then you can use $B$ as "easier" equivalent model.

What you actually do is to trust in the nescessairy math, that means if the axioms are contradictory (maybe choice axiom is wrong? :-) ) you make a mistake. What you do not need to do is to interpret each step for your model. I mean of course you can do that and the interpretation will fit to the model if done correctly but that might get difficult.

All you need is a "correct interpretation", that $A$ is a statement about numbers, (that means a mathematical statement) and that $A$ implies $B$ (the proof itself is unimportant, it does not affect the property that the models are equivalent)

it's all a bit vague... and full of opinion

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"What you do not need to do is to interpret each step for your model. " This is the fundamental issue. How do we know we dont need to do this? –  MSEoris Feb 23 at 17:42
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I'd say because we trust in the validity of a statement about numbers: $A\Rightarrow B$. In my opinion it's unimportant to interprete the intermediate steps because that won't affect the result. –  Max Feb 23 at 17:52
    
Sorry,this is not related but the word you're looking for is "contradictory", not "contradictous". Although "contradictous" sounds awesome and should definitely be made a word! –  rocinante Feb 23 at 17:59
    
thanks, correctizationed it :-) –  Max Feb 23 at 18:15

How am I to rest assured that these abstract manipulations aren't fundamentally changing things or introducing new nuances that I need to care about?

You don't "rest assured" at all. In setting the model, you already set some constraints already - constraints that by definition make your model < 100% depiction of reality. The point of the model is not to reproduce reality, but to create enough of an abstraction to make the problem solvable and also make it applicable to real life when things do change in real life.

As you are doing the math, you're not fundamentally changing things or introducing nuances at all. The math is just computation. Just because you may be ignoring some value from step X to step Y doesn't mean that you're changing things. The basis of ignoring some value usually rests on the fact that the value goes to zero or is otherwise negligible to the result and therefore can be safely ignored.

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Let me clarify a bit, my issue is that the actual manipulations of the equations dont have the direct coorelation to the physical problem that the initial setup does. This issue only gets more confounded with more of the manipulations. Think of representing some system as a matrix and then performing a SVD decomposition. what did you actually just do to your problem? –  MSEoris Feb 23 at 17:41
    
The mathematical manipulations have no correlation to reality at all, and neither does the initial set up. The abstraction of a system as a matrix is not a 100% reproduction of reality. SVD has nothing to do with reality because it is computation. SVD has nothing to do with reality because it's not supposed to. SVD is a computational tool, it is not a model. –  rocinante Feb 23 at 17:52
    
The result of your SVD is interpreted as solving your problem because that's how you set up your mathematical model. So I am a bit confused here why you think that SVD is supposed to have some correlation to reality when it is merely a tool to solve a mathematical problem. –  rocinante Feb 23 at 17:54
    
I mean it sounds that you are complaining that your calculator is not a shovel. Of course a calculator is not a shovel. But your calculator tells you how deep and wide you have to dig with your shovel in real life. –  rocinante Feb 23 at 17:55
    
Because often models start out as equations, then get converted to matrix form, then upon which we execute things such as SVD. How does this process affect the original model. Yes its a math tool, but whats it doing? –  MSEoris Feb 23 at 18:04

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