Mathematics: Truth or Tool?

This is a kind of appendum to the thread I made on the unified theory.

It was always my understanding that mathematics was a tool to be used to confirm what we believe to be the truth where it is applicable, and to make sure that conformity is met.
Planetary orbits are plotted, distances measured, designs passed from architect to builder etc etc...

Yet now it seems that if mathematics says it is - then it must be so. It has become the truth of the thing.
And the scientific community seems to be embracing this. Does being the fact it looks good on paper make it so?

I note that in most ATM discussions the first cry is - "Show us your equations!"
But if the equation for the multi-verse was shown would everyone go "Ok, fair enough, it must be mainstream"

Maybe I'm just seeing the thorny bit of general scientific views here, or being a stick-in-the-mud and having my own views on the thing, but I would be interested in BAUT member's feedback on how they feel mathematics should be used, and how far you can take it.

Is it just a tool, or has it become something almost akin to a deity that we most follow without question?

__________________
I don't believe in mathematics. Albert Einstein

Biologically speaking, if something bites you it's more likely to be female. Desmond Morris.

Quantum analysis is scientific dithering

Professor Frink: My observations n'hey, n'hey, show the universe could be a torus Weh, uh, or toriod it may like the typewriters and bananas and the monkeys with big teeth the biting the screaming Mm-hai!

Homer: mmmmm... doughnuts!

It's a tool, but it's an essential tool. The numbers are measurements and models for how things work. If you don't have numbers, your idea cannot be tested.

__________________
Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

There are a lot of things that can be tested without numbers. I don't think Darwin required numbers to demonstrate natural selection. Though yes, for cosmology mathematics are often important for making ideas testable.

I think the issue is more like this: knowing the math often makes the proposition testable, like Gillianren says, but also it is a demonstration that the person has an idea of what he/she is talking about. There's nothing godlike about mathematics, but being able to make something into a formula shows that you have thought through ideas. In a sense, it makes you a member of a club of people who have an understanding that makes your ideas more serious in a way.

__________________
As above, so below

I think this quote is as good an answer as any, from Zenos paradox (post No. 68).

Mathematics is necessarily true, but only to the extent it deals with mathematical objects, which are conceptual. There is potentially a difficulty in mapping the real world to mathematical objects.

So here I have "three apples", and I map that to the mathematical object 3.
And there I have "four apples", and I map that to the mathematical object 4.

It is objectively and necessarily true in the mathematical world that 3 + 4 = 7.

I map that 7 back to the real world and deduce that I have "seven apples".

Are we allowed to do that? Now it may seem utterly obvious that to the extent that 3+4=7 (and while that looks fairly simple, many other mathematical truths, while much harder to prove, like Fermat's last theorem, are just as utterly true), then 3 apples + 4 apples = 7 apples.

But 3ml of water + 4 ml of pure alcohol doesn't give 7ml of diluted alcohol. The "3" in 3ml doesn't map so well to the 3 in 3+4=7 as it did with apples, nor did the + work so well when we mixed the liquids. Even if did it as 3g + 4g and weighed the result with sufficient sensitivity, I would actually find a very small mass change (teh very fact that I call it a mass change implicitly assumes that addition works for adding masses of substances).

We do an awful lot of mathematics, geometry and more, in Euclidean space. This admits unlimited division, which doesn't match very well to what quantum mechanics tells us. It is sort of assumed that if actual space is "flat", then it is Euclidean. What about those volumeless points, lines and surfaces that geometry deals with? So there are quite tricky issues of the extent to which we can match the geometry we do in that theoretical construct to the actual physical space we have.

Even if I am confident of the mapping, like I might be with money and arithmetic operations (but I think that is begging the question, I think the way we use money we actually take it as the mathematical object, especially once it gets into banks as virtual money rather than being in our hands as notes and coins), I think it is an open philosophical question whether mathematical truths can be mapped back to the real world.

When we ask an ATM proponent to "show the math" or "show the equations", it usually is because the idea at hand can be tested mathematically, often at relatively elementary levels.

Last year someone argued that a cluster of marbles dropped from a very high altitude would expand and become less dense upon falling. I was able to show by means of some rough and dirty pre-calculus that the cluster would become more dense in spite of being vertically elongated. In another thread he argued that the observed distance/redshift pattern could be explained by assuming that the galaxies in question were falling toward a gargantuan black hole. I showed by means of relatively elementary geometry that such a model could not fit the observed data. He either did not understand the math or had some non-mathematical motive for rejecting my responses.

As Ivan said, Mathematics is a tool which can be used to argue about the world, but to do so and to know the value of the conclusions you have to be careful about how it's done.

You can map the physical world to a mathematical model and make lots of provably true calculations about that model.
The problem is that unless the mapping is correct, the conclusions made by those calculations won't map back to the real world.

Ivan's example of mixing water and alcohol is an example of what can happen if the model is wrong.
Model assumption: mixing liquids can be modeled by adding their volumes.
Real world: Two beakers with water and alcohol.
Mapping: We look at the water and finds it reaches the 3ml mark, this we model as 3ml of water; we similarly measure the alcohol and model it as 4ml of alcohol
Mathematical operation using the model: We model mixing them, so 3ml+4ml=7ml
Mathematical result: 7ml of diluted alcohol.
Mapping back to real world: If we mix those we should get 7lm of diluted alcohol.
Now comes the science part: we try actually mixing them, then measure the result.
Oops, there is less than 7ml.

This does not mean the mathematics are wrong, it means our mapping from the physical world to mathematics was wrong.
In this case by inadequately modeling the result of mixing two liquids.

Mathematics can never show anything to be true, it can only show the expected consequences of our models, which can then be tested against the real world to see how well the model fits.

Though it's not often explained that way, the main part of learning basic physics is to learn the terms commonly used for this mapping and the basic models used.

__________________
An emperor without enemies, a king without a kingdom, supported in life by the willing tribute of a free people.

Mathematics is amazing in that you can take some axioms that appear to hold in practice, do a lot of stuff on paper involving manipulating symbols without actually touching what you are modelling, derive some theorems, go out and measure things, and find out they agree with the theorem to many decimal places. Some author called it "the unreasonable effectiveness of mathematics". An example, take a rectangular patio. measure the two sides, square the lengths and sum them, then take the square root. A mathematical theorem says that this will be the diagonal length. Measure it, and find that it is right. In fact, if it doesn't come out right, it means you screwed up measuring!!!

Yes, if you use a poor-fitting model, it won't give the right results. But the fact is, good-fitting models have been found for so many things--Newton's "incorrect" mechanics is still good enough to put a man on the moon. The mathematical consequences of Einstein's general relativity give precise enough answers that GPS works--and we know this is not a complete model either because it contradicts Quantum Mechanics which also gives very accurate answers in its domain.

Roger Penrose suggested that mathematics is "unreasonably effective" because there is a "platonic world of mathematics" that exists independently of human mathematicians (i.e. if aliens did math, they'd come up with the same stuff, though with different names of course!), and that the universe's fundamental laws, whatever they are, are mathematically consistent.

Oh yeah, Hardy's own number theory research is today used in cryptography, so his ideal (a pun, an ideal being a mathematical object used in number theory) of non-usefulness didn't pan out even for his number theory work!

__________________
-----
Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

The danger is in when mathematics become 'elegant and beautiful', for then they take on 'godlike' properties. Cosmology, as a mathematically based understanding of the distant cosmos, is especially suceptible to this fetish approach to godlike mathematics. After all, what are we studying the cosmos for if not to find the truths of our universe... ie., default 'God'.

__________________
Credibility is simply incredible... sometimes even to me.
disclaimer

It is a tool. If your assumptions are garbage, the final result garbage would be.

__________________
"Shut up and calculate" R. Feynman

Not just some author. That was Eugene Wigner, in "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", written a few years before being awarded the Nobel Pirize in physics.

He wrote:

"The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."

and

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

Fadingstar wrote:

I think Wigner argues very well against this idea in his paper.

Population statistics are numbers. Beak size measurements are numbers. Years are numbers. I joke and say that the reason I took biology and geology for college math is that there's no numbers involved, but it isn't true. The ratio of predators to prey (1:40, I believe, but I could be wrong) is an equation of sorts. I actually had to do by-Gods equations in my oceanography class; we studied tsunami, and we had to work out the length of time it would take for them to travel x distance--and those numbers work.

No, Darwin used numbers, too. He just didn't need calculus for them.

__________________
Gillian

"Now everyone was giving her that kind of look UFOlogists get when they suddenly say, 'Hey, if you shade your eyes you can see it is just a flock of geese after all.'"

"You can't erase icing."

"I can't believe it doesn't work! I found it on the internet, man!"

At a minimum, you need statistics to determine if something observed is significant, or more likely explainable by random chance.

Todd

__________________
-----
Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

When I read the OP I thought to myself I’m going to reply and point you to Roger Penrose’s “Road to Reality” book, the first chapter; but tdvance beat me to it.

I would add to tdvance’s post that Penrose also talks about “The Mental World”, this is the picture you have in your mind of how the “Physical World” works and Penrose separates it from the “Mathematical World”. The aha for me was the separation. I build models that fall in the “Mental World” and the struggle for me is building the “Mathematical World”. Don’t get me wrong, I feel the “Mathematical World” is important; it’s what gives the “Mental World” or mental image a chance of surviving.

One of my favorite quotations is Einstein’s:

Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavour to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears it ticking, but he has no way of opening the case. If he is ingenious he may form some picture of the mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison.

For me thinking of a closed watch (being an engineer) I picture in my mind a set of small gears and a mechanism for turning them. I know the math! I can do all of the calculations to make the hands move precisely. I have the “Mental World” and the “Mathematical World” where they can exactly duplicate the “Physical World” exactly. But what does all of this mean if the closed watch is a digital watch?

Mathematics is a tool.

Jim

From Torsten’s referenced paper:
More to the point, these are axioms chosen in response to conditions demanded to solve a particular problem, whether in the natural laws of physics, or probabilistic conditions observed. Once these axioms are correctly chosen, the rest of the math, which is nothing more than an ‘interrelationship’ of how these axioms interact, is useful in defining a solution, which when tested against real events gives predictability of what the solution should be. If the predictability fails, then the math may be inadequate, which may either be because axioms chosen are wrong, or the interrelationships defined are wrong. But when they are right, and predictability is achieved, then the ‘magic’ of math becomes evident. Though the math may be perfectly self consistent per axioms chosen and how they subsequently interact, it is not a language of the natural world unless it passes the test of predictability.

IMHO, where math and reality may become confused is when we use a limiting factor to our axioms, such as the 'light speed limit' of our observations (a reasonable axiom), to qualify the interrelationships that occur from this limiting factor. This is what Relativity is all about. But if the limiting factor applies only to our observations (we see with light), axiomatically, but not to how the observed interactions interact in and of themselves (which may be faster than light), then our axiom merely limits our ability to observe things in and of themselves, but not necessarily from their own intrinsic point of view: Viz., an event at time zero, and zero distance, will be different from the same event seen at a distance over time, since it is not limited by light c.

This, in effect, highlights an axiom of Relativity, that there are ‘no preferred reference frames’, which then limits our ability to understand the interactions without such light speed limitation. So from our local point of view, we are limited observationally, but this does not mean that (at a distance) the events under observation are constrained by our light speed limit. The end result is that we then are left with a light distorted view of the world, where natural laws interacting within themselves may not be constrained by the light speed limit; but we per our axiom of light speed limit c are constrained from seeing it, so we do not see it as it really is, merely as we can observe it to be. If so, the math may be ‘beautiful and elegant’, but it may miss the point of what is actually happening, because of the axiom chosen. And if this is so, then that axiom of light speed limit c is not the right tool mathematically to fully understand nature on its own terms, if it interacts faster than light. (Of course, it is totally the right axiom if the universe interacts at light speed c only.) And if this is so, we have the wrong tool to understand the interrelationships of physics, except as limited by our observational limit; though what we observe is correct as an observational artifact, since we must use light or electromagnetic energy to observe phenomena at a distance, which is therefore corrected by the Relativity effect of light speed c. Taken outside of its ‘domain of applicability’ the result of observation may be different from the facts involved, if they interact at above light speed!

The only way to know if our observation is true or not is then to test it for predictability: Observationally, it will prove correct within its ‘domain of applicability’ and line of sight; but realistically, it may prove wrong locally (at time and distance zero), though we cannot know it! Why? Because our axioms chosen would not let us know we are wrong, since we chose axiomatically that there are ‘no preferred reference frames’ within the ‘domain of applicability’ for observations using light. But this may be wrong, since it assumes that what we observe at a distance is what is being observed locally. This is true, but only if one assumes that there is a ‘preferred reference frame’, that of the observer, since the universe may work above the light speed limiting factor (and not the other way around). Circular reasoning then takes effect, where we are proven right within our domain of applicability, but only observationally, and not necessarily true for the reality observed, which may break the axiom. Using the light speed limit c then of necessity gives us a limited understanding of what nature is really doing, if so. To break out of this circular reasoning can be done only one way, and that is with independent observation at a distance, where time and distance are both zero. But in astronomy this is impossible! So there is the conundrum for using mathematics as a tool of astronomy, which may prove correct observationally, but we do not know it to be true in fact. What we think we see may not be what is there. In fact, our ‘act of observing’ skews the result, just like in the Quantum world, because we are limited by the speed of light in all our observations.

And if distant aliens were to do the same mathematical observations they would come up with the same results, from their point of view within their ‘domain of applicability’, but with different units of measure, perhaps? Unless... they do not use the light speed c limit in their mathematical modeling… different mathematical tools... they'd come up with a different 'Arelativity' model.

__________________
Credibility is simply incredible... sometimes even to me.
disclaimer

Actually, the light speed limit is not an axiom.
If you do the maths, it's a consequence of the rest.

__________________
An emperor without enemies, a king without a kingdom, supported in life by the willing tribute of a free people.