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?

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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.

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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.

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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.

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‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

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!

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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'.

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

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

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What brings us together is stronger than what pulls us apart

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.

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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

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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

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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.

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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.

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‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

Even when theoretical physics is neatly tied up into an axiomatic form, the axioms are normally suggested by experience, or at the very least by physical intuition, and in any event they remain subject to future revisions, should any evidence arise that contradicts them. Pure mathematics (though the boundary can be fuzzy) is not like that. In pure mathematics, physical plausibility is a bonus, not a requirement.

So, definitely a tool, in so far as it's employed by physics and other natural sciences.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

Einstein's second postulate, the light speed constant in all reference frames, is 'axiomatic' to Special Relativity. No complaint on that, merely stating the obvious. The math works, as you say.

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

I think it's a tool (e.g., mathematical symbols) used to find truths (1 = 1).

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"Call me old-fashioned, but I think fire is magic. And it scares me a lot."

--The State

Having light speed constant (which is an observable fact) is not the same as axiomatically saying light speed is the maximum speed.

That it IS the maximum speed follows from the constant speed but is not an axiom in itself.

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‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

Then Henrik, you might like to join this BAUT discussion on FTL. That "it IS the maximum speed" is Special Relativity's axiom, based upon Einstein's postulates. Whether or not entanglement violates c is another story, and another discussion.

As someone said earlier, "garbage in garbage out", holds true for all axioms.

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

It's my impression that the speed of light being maximum is derived from the speed of light being constant, in which case it isn't an axiom.

It is admittedly quite a long time since I looked at the mathematics of it so I might be wrong.

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‘To those who regard “crime fiction” as some sacred icon which must follow a rigid formula, I will always be the man who writes 18-syllable haiku.’Trying to make sense of computers, The Error Log.

One thing that hasnt been discussed here, and one of the things that should show that scientists arent thinking that math is a god is that even in physics, the math can be complete BS.

(it has been a few years since my ED class, so I may butcher this a bit. For reference, all this is correctly done in Jackson)

If you do the math for radiation emission from an accelerating charged particle there are two solutions, +-d/c time from the current position of the particle. d is distance to particle from the point in question, c is speed of light. One solution is the expected delay in info transmission due to the speed of light, but the other solution, which is perfectly allowable mathmatically is the emission from where the particle hasnt gotten to yet. It is mathmatically an acceptable solution that kinda violates causality. It is therefore discarded.

Math is the tool we use to put science together. It allows many disparate things to be compared, and it allows the reasonable results to be found. The nearly universal applicability of statistics is due to the math.

This is also why we demand math from the ATMers. If the math is given, it can be tested and the limits of possibilities can be found. Frequently other tests of an idea can be found also. It gives a way to tell wether an idea is sound

I believe this is the case--Einstein's Special Relativity had two postulates:

1. there is no preferred inertial frame of reference
2. the speed of light is a constant

From this, it can be derived that nothing below the speed of light can be accelerated to above the speed of light, and if a signal could be sent faster than light, there would be causality violations.

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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

Perhaps an important issue to bear in mind, when we do observations, is that we are in effect reversing the logic that is used in mathematics. In mathematics, you choose axioms and apply the reasoning techiques that mathematics provides to find all the ramifications of those axioms, called "theorems" or more loosely "truths" (though the latter usage is imprecise and leads to conceptual problems that have come up on other threads about Godel incompleteness and complete unifiability). But in science, we generally start with observations, intepret them as clues toward theorems, and try to anticipate the axioms that could lead to those theorems as elegantly and parsimoniously as possible. There is a direct mapping from the axioms to the theorems they produce, but the inverse cannot be said-- any finite set of theorems may come from different possible axiom sets. So even when we find axioms that give the theorems we want, we still have to do more observations to see if those axioms also lead to additional theorems that work too. Amazingly, we often get a vast array of observationally correct theorems, when we start with just a few and reason back to "good" axioms and then find the ramifications of those axioms. I think this is what Penrose and Einstein and Wigner were all talking about as mentioned above-- why do a few good theorems and a savvy axiom choice lead to such a wealth of truth? It tends to make us forget that truth does not come from mathematics, provability does-- truth comes from observations.

When we get reminded of the difference between truth and provability is when we notice that the axiom sets we choose in physics actually lead to contradictions. This is not rare, it is quite common. But it doesn't bother us, for we learn to recognize which axiom subset will work best in any particular situation. Then we divide the axiom subsets into "courses" and "textbooks", and as long as we restrict attention to the appropriate subset of the observations, we are fine. Unification is always an effort to smooth over the cracks between these axiom subsets, and in another amazing development, we often find that this is possible to do to some degree. But again the tendency is to mistake consistency for truth, and more to the point, vice versa. There is actually no guarantee that the truth will be describable in a fully consistent way.

That's how mathematicians often proceed in their creative phase, too. Make observations, then infer a pattern.
But mathematics allows for an extra stage in the process, where you get to follow the route in reverse, and prove beyond any rational doubt that your inference is true (within a certain set of common assumptions). This is not possible in the experimental sciences, where the best you can do is gather more and more evidence in favour of your "theorems" (principles, or laws, or theories, or whatever you choose to call them), but your conjectures always remain subject to some doubt, well supported but never absolutely proven.

Here I disagree strongly with you, as you know. I find your concept of truth too restrictive.

(1) Make observations --> (2) infer hypothesis --> (3) test hypothesis --> (4) axiomatise hypothesis --> (5) derive other consequences from axiomatic --> (6) confirm consequences empirically

Why does this strategy always work so well? You say "because it's grounded on observation." But there is no reason why one observation should imply or justify another. I say it's "because it's grounded on mathematical derivation, which follows the rules of logic." I would be very surprised to see the (4) - (5) - (6) part of this chain fail to work, unless the axioms in (4) turn out to be wrong.

I suspect that you're using the word "contradiction" in a very loose sense, there. I've never met a real contradiction in science, only things that seem contradictory at first glance, but stop being once you analyse them with more care.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.

True enough-- mathematicians of the "non-Hardy" flavor. There is indeed a tendency in mathematics to start with some familiarity, work backward to the axioms that give it, then work forward from those axioms to see how you did-- much like in science. But if you started with a familiarity based on observation of reality, you are in effect doing science not pure mathematics. The distinction isn't really between "physics and math", it's more between whether you start with axioms you like, or start with familiarities you wish to make contact with. What ends up happening is often a kind of combination, so the distinctions are not so cut and dried as I may have suggested.
Yes I agree, you can prove that a certain prediction stems from a certain axiom choice, and you can never prove the prediction is correct without testing it. In practice, we consider "good enough" to be any test under conditions that you deem "similar", but this "similarity assumption" is yet another way in which science is not mathematically precise or unifiable.
The truth I refer to is not meant to be an absolute concept of truth, as I've said in the past it is strictly the concept of "scientific truth"-- the objectively repeatable footprint left by some far more inscrutable beast that science has no way to define or address. I'm not sure if you are objecting to my ruling out of other forms of truth, which I did not intend (truth vs. scientific truth is another thread), or if you feel that scientific truth can be extended outside of what is observable.
But the process (4)-(5)-(6), those "rules of logic", were also built on the backs of observation. Your position only works if you view those rules as separate from the axioms, as though given to us separately from the process of choosing axioms. I view the rules of logic as just more axioms that we chose because they work, just like the other axioms. We are not born with the ability to be logical, it is our experiences, our observations of reality, that train us to do it.

All our minds can do is organize our familiarities, the idea that we can do something more fundamentally separate from reality is hard to support. Even if one attributes "instinctive" logical capabilities to our minds, they would have been "chosen" by natural selection, i.e., chosen to conform to the observational tests of who survives. If we count "death of an illogical brain" as a kind of "observation of how reality works", then it's still all grounded in observations.
Many people seem to think that, yet contradictions abound. In practice, what you mean by "stop being contradictions when analyzed with more care" is just saying that "we learn how to cleverly sidestep the contradictions so that we still achieve our goals". Classical mechanics talks about the location of a particle-- that contradicts quantum mechanics. Yet classical mechanics is still here, and that axiom set is still chosen, more often than the quantum one. Some might say that the contradiction is not fundamental because we don't have to use classical physics, but in fact we do-- you really can't treat a baseball without making idealizations that are tantamount to doing classical physics. The point is, you always have to include idealizations, you just get to choose what they will be, based on how well you deem them to "test out" in the end. Contradictions at the axiomatic level are therefore of no different status than contradictions introduced by the idealizations needed to apply that axiomatic subset.

Then of course there's the usual contradictions between general relativity and quantum mechanics, on the more fundamental level. Much is made of this latter problem, but I see contradictions all over the place in physics-- like the need for boundary conditions to make solutions "physically meaningful". Where is the "theory of boundary conditions"? There isn't one, we use whatever works. That is a contradiction to the idea that we are using an axiomatic system, because contradictory predictions are made by different boundary conditions, and often the only way we know which boundary condition was most appropriate is by looking at the tests. The predictability then relies on faith that any other situation will require similar boundary conditions.

Paul Davies covers this briefly but rather deeply in his 2007 book Cosmic Jackpot (see ~p.235-9). I can't cover all the points made here, but he starts out by saying....

Most theoretical physicists are Platonists in the way they conceptualize the laws of physics as precise mathematical relationships possessing a real, independent existence that nevertheless transcends the physical universe.

Of course, he adds....

Many physicists who do not concern themselves with philosophical issues prefer to think of the laws of physics more pragmatically as regularities found in nature and not as transcendent immutable truths with the power to dictate the flow of events.

Davies refers to J.A. Wheeler quite a bit in this book (well, it's dedicated to him), and Wheeler was "perhaps the most committed anti-Platonist." Wheeler liked to quip, "There is no law except the law that there is no law."

One concept I kind of liked was Wheeler's idea that the laws of physics did not exist a priori but emerged from the chaos of the quantum big bang, "congealing along with the universe that they govern in the aftermath of its shadowy birth." That is, they emerged in approximate form and sharpened up over time. Obviously, then, the mathematics could not be an expression of eternal truth.

Then there was this other point: Newton/Leibniz came up with calculus, which requires variables to vary continuously. You have to assume space and time are continuous and infinitely subdivisible on any scale of magnification, right down to zero.

Bottom line:

Platonic laws can perhaps be treated as useful approximations, but they are not "reality." Their infinite precision is an idealization that is normally harmless enough, but not always. Sometimes it will lead us astray, and never more so than in discussion of the very early universe.

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Everyone is entitled to his own opinion, but not his own facts.

That is an interesting idea, but if it's true, one wonders, but what about the laws of the chaotic system from which the laws of physics emerged? After all, there is plenty of literature on the mathematics of chaotic systems! That something orderly could emerge from it strongly suggests it's not "pure randomness".

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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

Does not follow. We now know that Newton's physics does not work "on any scale of magnification, right down to zero." It's merely a (very) useful approximation for macroscopic scales. Below that, you need quantum mechanics.

If classical physics is only supposed to be an approximation that gives good results in some conditions, but not in all, then there is no reason why the mathematical assumptions of calculus should represent everything in the universe "literally", "right down to zero." It would be a straw man to claim (if that's what Wheeler was suggesting) that the discrete, atomic nature of the universe (but is it atomic and discrete still in QM?) contradicts calculus. Because the parts of physics that use calculus are not meant to be exact representations of the universe in the first place, just effective macroscopic models.

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"A witty saying proves nothing" Voltaire.
"All your bias are belong to us" Ara Pacis.