Exploring Math and Assumptions in Our World

[Digit1]

So lately I've been digging into geometry and I always seem to come back to the same question, how can we really test to make sure our math values can correctly apply to world that we live in? First, let's assume that we live way back in time and our scale of measurements where limited (for example, we couldn't go past 1 decimal point) For example, if we construct a simple wheel (lets say with a measurement of "1") and take a piece of string, wrap it around the wheel and make a cut where the string just starts to overlap. We can measure that ~3.1 units long. Now, if we apply the same concept in math we end up with an irrational number 3.1415... So, as our scale of measurements got better and allowed us to measure things better, we can measure out to more decimal points. Now, say we can measure out to 2 or 3 decimal points. Where is the point that someone said "Alright, the numbers are close enough. I can for sure say that in the world we live in, the ratio of the circumference of a circle to it's diameter is equal to pi." It just seems like it would be a huge assumption to make back in the day. Again, we take it for granted now, yes we can take really really really small measurements, but we can't be 100% sure. Always bothered me a bit. Kind of a side note,is it possible that our number system won't hold up in the future? For example, when I learned a bit about quantum physics you can have a particle be super positioned and we found out that observing particles will change their behavior. Is it possible that our whole math system is becoming "out of date"? Or does the language need to be expanded?

Oh and another thing, it seems like in order for us to stumble upon math and understand it, we first needed to apply experiments in the world we lived in first. After we got a good grasp of things, we then didn't need physical models to test things out, math was self supportive, or so I believe. I'm just curious if recently (within the past 50 years or so) if we had to do physical experiments before a new branch of mathematics where created. It seems that at first applied math was a must, but then we where able to discover and practice theoretical math more.

Thanks.

 

[Pengwuino]

I suppose the best answer to your first question is that we've never made a measurement where the value of pi was anything but 3.1415... and so on within margin of error. We obviously have various non-experimental methods of finding the value of pi and we know the value of it to a ridiculous number of digits. Experimentally though, there's always error involved and you'll never know exactly what pi is experimentally (of course, since pi is irrational seemingly, there is no exact answer in the first place).

As for the QM example, how does a particle being in a superposition of states have anything to do with the mathematics being faulty? Quantum physics isn't a result of the math, it's the other way around. The math, namely linear algebra, is used to describe quantum physics. The "weirdness" people typically associated with QM is simply because that's how the world works, not because the math is faulty.

 

[CRGreathouse]

Oh and another thing, it seems like in order for us to stumble upon math and understand it, we first needed to apply experiments in the world we lived in first. After we got a good grasp of things, we then didn't need physical models to test things out, math was self supportive, or so I believe. I'm just curious if recently (within the past 50 years or so) if we had to do physical experiments before a new branch of mathematics where created. It seems that at first applied math was a must, but then we where able to discover and practice theoretical math more.

I think the usual (but not only!) direction is the reverse: the theoretical is developed first, then an application is discovered.

 

[Tibarn]

If you draw a big enough circle on Earth, you'll find that the ratio of the circumference to the diameter begins to deviate away from pi. Math is concerned with what you can deduce from certain assumptions. Applying math to the real world strongly depends on which assumptions hold. Essentially, the real world is a special case of abstract mathematics. Applying abstract math to the real world depends on which case it is, and that's a problem for physicists to figure out.

 

[Pengwuino]

If you draw a big enough circle on Earth, you'll find that the ratio of the circumference to the diameter begins to deviate away from pi.

... because the Earth isn't perfectly spherical.

 

[Digit1]

Hm thanks for the responses so far.

Tibarn: Where you referring to how 2d euclidean geometry doesn't hold up if applied to 3d space, where space folds and whatnot?

Pengwuino, when you stated `The "weirdness" people typically associated with QM is simply because that's how the world works, not because the math is faulty.` I was curious as if their is any mathematical way that this weirdness can be explained. For example, when complex numbers where found it allowed us to solve many problems that seemed impossible to be solved at the time. Just curious if math will be expanded to the point where somehow conscious thoughts can be expressed and somehow can be used in calculating values. I know this sounds a bit weird and maybe even stupid but we couldn't get to where we are now withought asking questions =P.

I just find it a bit funny how when I took calc it was one of the easiest math courses I taken in a while, but when I sit down and try to re-construct math and understand more simple concepts, a lot more questions arise. Basically I try to take nothing for granted when it comes to math, if I don't understand something I try to reconstruct the concept and see how it was found. Sitting down with a pen and a piece of paper and re-creating geometry isn't the easiest thing in my opinion =P.

 

[Fredrik]

how can we really test to make sure our math values can correctly apply to world that we live in?

We can't.

Where is the point that someone said "Alright, the numbers are close enough. I can for sure say that in the world we live in, the ratio of the circumference of a circle to it's diameter is equal to pi."

We never reached that point, and there was never any chance that we could.

Again, we take it for granted now,

We don't "take it for granted". We just know that these things can be proved to be true, given a specific set of axioms. Some of these statements may not be true if we start with a slightly different set of axioms.

...yes we can take really really really small measurements, but we can't be 100% sure. Always bothered me a bit.

It should, because you have probably been told at some point that a theory (of science) is a set of statements about the real world that are believed to be true. What you're starting to realize is that experiments can't ever prove a theory to be true. All they can do is tell us how accurate the theory's predictions are. What this means is that the naive definition of "theory" is extremely inappropriate (for physics, but probably good enough for other sciences). The best definition (or at least the main part of it) states that a theory is "a set of statements that makes predictions about probabilities of possible results of experiments". (In most theories, those probabilities are always 0 or 1. The only exception is quantum mechanics).

Note that this problem isn't a problem for mathematics. It's just a limitation of the scientific method. Science can't tell us as much as we would want it to.

Kind of a side note,is it possible that our number system won't hold up in the future?

It hasn't been holding up for the past one hundred years, if by "hold up", you mean "appears to be an exact description of the properties of space and time". (That's what all your talk about measurements suggests to me). The real numbers are defined to have properties that agree with our intuitive ideas about space and time, but the best theories of space and time are pretty non-intuitive.

For example, when I learned a bit about quantum physics you can have a particle be super positioned and we found out that observing particles will change their behavior. Is it possible that our whole math system is becoming "out of date"? Or does the language need to be expanded?

So far the math based on the ZFC axioms of set theory is more than good enough for physics. (This includes every part of math that you have heard of, and a lot more). If you meant something much more specific by "our whole math system", such as using [itex]\mathbb R^3[/itex] to represent space, we're already far past that point.

 

[dx]

The best definition (or at least the main part of it) states that a theory is "a set of statements that makes predictions about probabilities of possible results of experiments". (In most theories, those probabilities are always 0 or 1. The only exception is quantum mechanics)

I'm not sure I would include "probability" in the general definition of a theory of physics. I would say a theory is a mathematical structure M, together with a map μ from experience to M.

 

[Fredrik]

I'm not sure I would include "probability" in the general definition of a theory of physics. I would say a theory is a mathematical structure M, together with a map μ from experience to M.

That seems incomplete. Maybe you can elaborate a bit. Does quantum mechanics qualify as a theory under this definition? What do you mean by "experience"? Does the set of experiences have a mathematical structure too?

 

[Dragonfall]

If you count every ordinal as an "experience", then the class of experiences is actually a proper class under ZFC.

But then proper classes are experiences too... perhaps you need Russel's theory of types to rigorously define physics.

 

[dx]

That seems incomplete. Maybe you can elaborate a bit. Does quantum mechanics qualify as a theory under this definition? What do you mean by "experience"? Does the set of experiences have a mathematical structure too?

By "experience", I mean our raw sense-experience. For exameple, we intuitively map our visual experience into the mathematical structure E³ (Euclidean 3-space), and on the basis of this mapping, make certain predictions that allow us to navigate. In general, a physical situation is mapped into some mathematical structure, and then we use the concepts and constructions available within that structure to make predictions, which are then related back to experience.

Quantum mechanics is of course a more or less well defined mathematical structure, and we know (more or less) how to map experience into it, so it fits the definition.

Does the set of experiences have a mathematical structure too?

I would say the goal of physics is to find the mathematical structure of the set of experiences.