... or the patio is not quite rectangular (corners aren't right angles);
... or the space on which you measured it isn't Euclidean, say your (humongous) patio covers a substantial part of the earth's surface, or is on asteroid Philplait!

After reading through the posts, I think that the general consensus is that mathematics is a tool, and not something to be expressed and taken blindly as truth. On the ATM bit, I agree that equations should be shown where they can, and used in argument against.
I didn't in my ATM post because I was more concerned with the idea of the thing (sort of philosophizing) rather than the mathematical model of it, which kind of lead onto this thread.
One thing I couldn't visualize though was a world without mathematics. I gave it some thought and got us back somewhere around the Stone Age - then thought Flintstones (haha) - but even they must use mathematics.
O yes, it's a sorry state of affairs when you're considering the plausability of a cartoon!
But, seriously, the mere fact of counting would be considered a mathematical advancement, so perhaps pre-Stone Age would be more appropriate.

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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!

To me, the issue is not whether the "laws" emerged slowly from other laws, if we are to presume that would require having other laws about how laws emerge. It is how well can we rely on the language we use about laws "governing" behavior in the first place. I think this is a shortcut, that often works fine, but what evidence do we have that this is the reality? When we find a "law" that works, we tend to say it "governs" reality, just look at the language of "laws" and "governors"-- it's classic personified imagery (we interpret according to our familiarities). But then when its limitations are discovered, we conclude "it was only approximately the right law, there must be a deeper law that is the real one". As if we found a Lieutenant but still seek the General. Why would we not simply conclude that we have noticed patterns in past behavior that is predictive of future behavior? The elegance is in organizing the patterns in the most mathematically clear way, not in finding why reality does what it does. At what point are we not organizing patterns in our familiarities, but instead figuring out what makes reality work the way it does? I might go even farther than Wheeler: never. It's just not what we're doing, it's only what we're pretending to be doing.

But what's really amazing is that classical mechanics itself makes the assumption that infinite subdivision is possible, that's how it connects to calculus, yet we know this is not true, as you say. So the surprise is, we have a theory which requires an assumption that isn't true, yet the theory works anyway as long as you don't look too closely.

This is why I feel the profundity of Zeno's paradox with Achilles trying to catch a tortoise is not well appreciated. Many people dismiss it by saying "Zeno didn't understand that an infinite sum of smaller and smaller times can sum to a finite time". Maybe he didn't, I don't know, but the real point is-- we don't have physical access to that infinite sum of shorter and shorter times! Quantum mechanics would set in at some point if you actually tried to analyze Achilles catching the tortoise like that, and at some point the energy you'd need to see if that was "really happening" would fry both Achilles and the tortoise. So the infinite sum that calculus gives us is not "real", in the sense that it is not verifiable in detail with experiment, yet as an idealization of what is happening, it works great. In other words, classical mechanics is describable by mathematical idealizations that reality is not, yet its results describe the reality. I think Zeno's paradox is alive and well, and shows why our "laws" are not "what reality is actually doing", but are "mathematically elegant ways to organize our familiarities". Why this works is the deepest mystery of them all, one we are nowhere close to "unifying" with anything we observe. It's really the question, given that intelligence is the ability to unify familiarities, from whence comes intelligence?

I'm not so sure that that assumption is truly necessary for classical mechanics. From the mathematical side, at least, much of what we do with the reals can be done as well (though with greater effort) with the rationals alone. This is the idea behind finitism. I won't claim to be well versed in this interesting but rather obscure field, but my point is that a lot of the mathematical results we think require that our variables be continuous actually remain valid (possibly with minor adaptations) without that requirement.

But I'm actually more interested in the physical side of the equation. What makes you think that classical mechanics implies or assumes an infinitely divisible world, from a physical point of view? Could you give a couple of specific examples?

Well, we currently think we know better...

I said almost the same a while ago here in the forum, in a different context. Still, perhaps we shouldn't be too surprised. You analyse a phenomenon carefully, and you attempt to identify its most important features. If you then use those features as axioms to construct a model for the phenomenon, there's a good chance your model will provide a fair approximation to the real thing.

Although the atom isn't really made up of little particles called electrons revolving around a nucleus in circular orbits, matter is nevertheless atomic to first approximation, and atoms do contain a nucleus with protons and neutrons, and much smaller electrons dispersed around the atom. So, while Rutherford's model of the atom is strictly speaking false, reality is close enough that it can't help behaving a little bit like the model. Perhaps this is simply a logical necessity.

Remember that Galileo, who founded the scientific method, also said that the book of the world was written in mathematical language.

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

The rationals are also infinitely dividable, it's just countable. But I think I take your meaning, and it's an important one-- that if we designate in advance the accuracy goal we desire, we can apply a form of mathematics that only uses a finite number of operations on the natural numbers to achieve that accuracy, and continuous calculus is just a more elegant form for obtaining estimates within that same accuracy target. In that view, we don't use continuous calculus because it is "closer to the reality", or even more accurate, but simply because it is easier.

But the problem is not that you have to use continuous calculus to do classical mechanics, because the equations could be translated into finite versions. The problem is that we cannot arbitrarily tighten our accuracy goal-- at some point the classical predictions no longer work. If we ask, at what time does Achilles catch the tortoise, we can only get the answer right to a certain accuracy, after which the question no longer has meaning, even though we tend to imagine that the time that comes out of the classical calculation actually means something other than just a continuous estimate of an arbitrarily chosen finite process. Classical mechanics does not tell you that limitation, you have to apply it externally to avoid false predictions. Yet classical mechanics works fine at lower levels of accuracy, completely oblivious to this catastrophic failure at higher accuracies. That this is not actually a catastrophe, but simply the way we have become accustomed to using physics axioms, is very much the point I'm making-- that we were never doing anything different, we just became aware of the pretense.
The dynamical equations are differential equations, so they presume divisability of space and time. I agree they could be written in a finite (and very clunky) form, but the choice of the intervals, and the loss of accuracy that results, would be entirely arbitrary. There would be nothing in the theory itself that suggested it needed to be written that way, we would simply be imposing it because we know that otherwise it makes assumptions that are unverifiable. We do it the way we do because we prefer inconsistency to arbitrariness, it's just that simple. Maybe also because the math is easier.

Yes, I agree-- things get less surprising when we step away from the idea that we are "unmasking reality" and just accept that everything we do is an effective theory. Somehow we have become masters of modeling a fake version of reality that matches the real one, in many cases, astonishingly well.

But it still doesn't seem likely that this would even be possible-- like the difficulties we have predicting weather, for example. Why is it that we can isolate the behavior of particles enough to ignore "the weather"? If we couldn't do that, we'd be dead in the water. Perhaps there's some anthropic way of looking at that, whereby a universe that is not understandable by intelligence does not evolve intelligence. In fact, that's pithy enough for a signature!
I really don't know-- it seems pretty mysterious. It's ironic, most people find it weird that classical mechanics breaks down in the atom, but I find it weird that it doesn't do worse.
Yes, famous physicists have a very uneasy marriage with mathematics-- the same ones will be found to make statements like that, then turn around and poke fun at abstract thinking!

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Modeling the universe with real numbers a-la classical mechanics: one reason we use real numbers rather than, say, million-digit rationals is you get a lot more complication with a "rational" model, but without any discernible improvement in accuracy.

On the other hand, you get a tremendous (in terms of what can be measured) increase in accuracy by switching to Quantum Mechanics. However, instead of infinite-precision reals, we now have---complex-valued wave functions! Not only do we go from finite precision to infinite precision, we go from 3 dimensions for a particle in space, 3 more for momentum, but we actually have an infinite number (worse, a continuum of) dimensions--a complex-valued wave function is essentially a vector of complex numbers of dimensionality equal to the number of real numbers. One might be able to come up with a Rube-Goldberg model that avoids the use of wave functions and complex numbers, but I doubt you'll get an improvement in accuracy for the increase in complexity.

The upshot is, mathematics wins again in the race to understand the universe. Concepts of mathematics that were discovered by pure mathematicians turned out to fit reality very well. General Relativity is another example--differential geometry and tensor algebra, and some other things, come together to make the most accurate large-scale theory there is. If Einstein hadn't had differential geometry available to him, it is very likely he would either have stopped at Special Relativity, or at best, come up with a patchwork Rube-Goldberg theory for how matter behaves in various situations--no gravity field, constant field, constant rotation, constant acceleration, rotation+acceleration, rotation+gravity field, and so on.

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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

You got things twisted around a bit, it's (1) Make observations -> (2) Infer hypothesis -> (3) axiomatize hypothesis -> (4) derive consequenses -> (5) device experiment to test consequenses -> (1).

You can't test without observing.

I would say it works so well, not because it's good at showing we're right, but because it's very good at showing us when we're wrong.

The fundamental axioms of science is that all phycisists are capable of selfdelusion and that any hypothesis may be wrong

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

What's interesting is, I agree with both these statements, yet I also see how they contradict each other. I find many contradictions are fundamental to science, which is why I find it so amusing that the "holy grail" of science is complete self-consistency. As I've said elsewhere, I view the search for consistency as being akin to using the North star to go north, not as a guide to walking to Polaris.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Never forget a simple truth:

Figures can lie because liars can figure.

I do feel that scientific truth may be extended beyond what is observable, or, as I prefer to put it, I feel that the concept of "observable" should not be restricted to "physically observable". The way you phrased another part of this post showed the crux of our difference of opinion so far:

You're using a notion of "science" that excludes mathematics (let's reserve that term for "pure mathematics", here), because math is not based on physical evidence. This is a common definition used by many people, but I wish to challenge it, because, in the context of these discussions we've been having, I am convinced that making such a distinction contributes more to cloud the issues than to clarify them.

The notion of "science" you used, while common, is not the only one. There are also authors, as you must know, who would include mathematics among the sciences. Although pure math is not subject to the veredict of physical evidence, I would argue that it is subject to a different standard of proof, which comes from elementary logic. I would also argue, as some authors do, that elementary logic is no less objective than physical evidence. In fact, even in the day to day practice of the experimental sciences it's often difficult to disentangle the one from the other, as observations are also to some extent theoretical constructs (you've agreed to this in previous discussions).

In short: I reject the dichotomy between objective natural sciences on one hand, and arbitrary theoretical mathematics on the other. I say that mathematics is every bit as objective as the natural sciences, except that it is grounded on a different kind of evidence (logical, rational, or intuitive).

I won't claim that this point of view is better than yours, or that I can prove it is truer, but I put it to you that this philosophy, that regards evidence and logic as the two sources of objectivity, is no less defensible than the philosophy that sees physical evidence as the only possible route to objective knowledge. I put it to you as well that, in any case, the latter philosophy is no more than a philosophy: a good scientist may accept it, but he is not required to.

Not every one agrees with that. Many thinkers throughout the ages have argued the opposite, that logic is indeed an innate ability of our species.

While I personally find very plausible the conjecture that much of our thought processes are a result of our biological evolution, I am not sure I would extend that assumption to our entire intellect, including basic logic. As such, for the purposes of this discussion, I will reject that extension, on the grounds that I have never been shown any conclusive evidence that the whole edifice of logic, from top to bottom, is merely a tool molded by our environment through evolution in a contingent fashion.

As an alternative, I offer a different conjecture: that the rational part of our minds is not a mere product of our environment, but rather the environment itself has been conditioned by the rules of logic, because we live in a logical universe. This is why we've been so successful at using reason to understand the world. It was not natural selection which chose our logic, but logic which chose our universe.

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

Right, pure mathematics is a kind of mental game. I do not mean that in any derogatory sense, merely that it is an endeavor that we made up the rules and carry out primarily for our own intellectual satisfaction. That such endeavors may later connect with other familiarities is the point where they make contact with science, not before. I'm just using the standard definition of science.
But what do you mean by "objective", if not "supported by observations?" If we use elementary logic because it is supported by observations, then we are doing science, because we are making contact with our familiarities. If we are using it just because it is a choice of how we wish to order our thinking, then we are doing pure mathematics. The line is gray because there are combinations, but the dichotomy is useful when the two approaches don't agree (the times we need to use "fuzzy logic", for example).
Yes, the two are deeply entangled, but one can still separate them in principle. The difference is essentially in the direction one follows. If one is interested in starting with reality and reasoning back to axioms that represent, unify, or describe some important element of it, while projecting away all that does not fit into the system of validation you are using, then one is doing science. If one starts with the axioms and reasons out the logical equivalences, there is no projecting occuring, and one is doing pure mathematics. The key point is that "success" in the latter endeavor is based in provability and consistency, whereas success in the former endeavor hinges entirely on practical usefulness so inconsistencies (in the complete process, not the "math subroutine") are tolerated constantly. Thomas Hardy's famous toast to pure mathematics is relevant to this distinction.
But that different grounding is no small detail (just look at how mathematicians test their intuition compared to how scientists do it-- forming proofs versus doing experiments), it is the core of the distinction I'm drawing. Nevertheless, I agree with you that when I refer to its rules as "arbitrary", I just mean that a mathematician can choose different rules and still be doing mathematics (like the various other "schools" of proofs). I don't mean to suggest there is not something special about logic, we certainly learn logic from experience and there is something logical about reality. But many people take that too far and conclude therefore that reality is completely logical, a strange conclusion given that it follows from no valid logical syllogism.

In summary, the distinction is that pure mathematics says "let's choose a form of logic and see where it gets us, clinging rigorously to all avenues where we are led", whereas science says "let's choose a form of logic with a good track record for mimicking reality, confront observations, and not worry terribly about esoteric cracks in the facade if we get the overall picture right". There is certainly a conversation between these approaches which has a lot to do with the logic we normally use in both. I think you are stressing the importance of that conversation, and I don't dispute that, it is very important indeed. But where you see the difference is when you encounter "cracks" in the logic-- inconsistencies don't derail science, but they are a catastrophe for what rests on rigorous provability. The difference is very much the distinction between a "proof" and an "explanation".
I think I see the problem here. I am not saying that the use of logic is not an inherent part of science, I am saying that the logic is chosen to get results that connect to observations. The authority in science is that it "checks out" when confronted with observations, while pure mathematics has no need for any checks at all. That means that if the scientist gets the result that works, that informs them that their logic was "good", and they might suggest that the mathematicians use that logic because it seems successful. But the scientist invents a concept that is alien to mathematicians: the crucial concept of "close enough for my purposes".

Pure mathematics could never work that way, it has to already take the logic to be "good", and see where it leads. True, science informs pure mathematics as to useful logical choices, but it leaves a lot of leeway for the fingerprints of the mathematician (constructive proofs, axiom of choice, etc.), and pure mathematics informs the "subroutines" of thinking used by scientists, but it leaves a lot of leeway for the fingerprints of the scientist (boundary conditions, simplifying idealizations, accuracy targets, etc.). So the distinction is important to make, even though one should not oversell it to the point of ignoring the important conversation between them. I think you see me as overstressing the former to the exclusion of the latter, with words like "arbitrary" that could mean "unconstrained", and I agree we should not do that.
I could think a capability, like the capability of swimming, moreso than an innate ability-- I think it is quite clearly trained, just as magical thinking is trained. No doubt our brains have the capacity for both in various measures, but our training determines which we rely on. Just look at a typical evolution vs. creationist debate. (Or the example I once heard of a young girl whose father told her that he has been Santa Claus all this time, that no real Santa came with toys, and she said "well maybe you brought those presents to our house because you didn't believe in Santa, but other kids who did believe got theirs from the real Santa. Imagine losing power over the concept of Santa Claus with your own daughter!) On the other hand, there are situations where logic may let us down, such as when we wish to explore aspects of life that may lead to greater happiness or a richer experience. In those situations, a certain measure of "magical thinking" may be entirely appropriate for bettering our experience (is there not a measure of magic in things like love, art, music, and experience itself, and that all of these concepts can be described completely in terms of logic is itself a form of magical thinking in my view).
And whether or not that's true doesn't really matter-- we are thinking beings who could choose to go against our evolution and use a different form of logic if it suited our needs better. It is those needs and the mode we employ to achieve them, not the logic itself, that makes the distinction between science and pure mathematics that I'm referring to.
Certainly all possibilities should be on the table, as each might have its lessons. The question you pose here is probably one of the toughest questions of them all-- do we lay logic over our universe like a template that works for many things, and obliterates what doesn't fit like the way a shadow obliterates the third dimension, or is the universe quintessentially logical and so everything that happens must fit into the "logic template"? I think it is more logical to adopt the null hypothesis, i.e., the former hypothesis, until one can rule it out with evidence. That doesn't mean we shouldn't entertain all possibilities, it just means that I see flaws in the "positivist" approach that starts from the assumption of a perfectly logical universe.

Godel's theorem is relevant here as well-- from it we know that there are truths about rich systems like the reals that cannot be proven from a finite set of axioms, and if we allow simply listing all that is true as our set of axioms, then saying "reality is logical" becomes a tautology. We don't know that reality has the richness of the real numbers, but reality conceived of the real numbers, so the concept is part of reality, and entails true concepts that are not logically provable. In short, one cannot equate in all cases truth with provability, unless one defines the axioms of the latter to be the former, which begs the question.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Going back to Ken's latest post, I won't do a "point-by-point rebuttal" in the style of discussion boards, but will attempt to address a few excerpts from your post which I think capture the the crux of our diagreements and misunderstandings.

Yes. The problem here is precisely that we are not using those two important words "objective" and "observation" in the same way. Let me address the former first.

What is objectivity? I would explain it as follows. When some activity is such that, given enough time, all rational and sufficiently informed people eventually converge upon an agreement about it, it is objective. This is what happens with the natural sciences. There was much disagreement at first about, say, global warming -- or continental drift, or quantum mechanics, to use less popularly controversial examples --, but eventually most scientists came to accept all three theories. There are, of course, various people who doubt each of them. But when we investigate the doubters closely we typically come to the conclusion that they are either not sufficiently informed about the theory in question (in which I include those who do not make the necessary effort to understand it), or are being swayed by irrational impulses.

Another important part of objectivity is of course that the debate must be mediated by some kind of evidence: climate, ice, and gas measurements and estimates in the case of global warming, geological and paleontological traces in the case of continental drift, sophisticated laboratory experiments in the case of quantum mechanics. That's the standard of proof we abide by in science. I would say that this is the basic formula of objectivity: rational consensus mediated through evidence. I hope you will agree.

Where we part ways is in what we allow to constitute "evidence". You read this word and are only able to think of physical evidence -- the "observations" you mentioned. But I argue that logic provides us with its own kind of evidence. Good ideas in science are logically consistent, and consistent with other good ideas. Inconsistent ideas are invariably bad; we don't even bother to test those out empirically, most of the time (see a recent example with some resemblance here). I say that this is evidence, too; evidence of another kind.

As I wrote before, I disagree with that philosophy. Against it, I would give two arguments, one of a rationalist nature, and the other of an empiricist nature.

1/ The rationalist counterargument is that the constructions of pure mathematics are not an arbitrary game, that could just as well have been constructed according to different rules -- the definition of a game is precisely that its rules are a matter of convention (2 + 2 = 5?) --, but rather follow rules of logic which rational beings cannot break without denying their own rationality. These rules are not merely a subjective part of how our brains are wired, we find them in the very architecture of the universe. You are free to disagree, but then the burden is on you to show how an illogical universe could exist.

2/ The empiricist counterargument is that, in separating "pure mathematics" from applied mathematics, you