Science: the toroidal pyramid

Chad Orzel gripes about this month’s Scientific American special issue on “The Future of Physics” — which is actually extremely good, but which turns out to be exclusively about the future of high-energy particle physics. Not surprisingly, the commenters on Chad’s blog reignite the ancient debate about which science is more fundamental than which other one, and whether all sciences besides particle physics are stamp collecting.

I started writing a comment myself, but then I realized I hadn’t posted anything to my own blog in quite some time, so being nothing if not opportunistic, I decided to put it here instead.

To me, one of the most delicious things about computer science is the way it turns the traditional “pyramid of sciences” on its head. We all know, of course, that math and logic are more fundamental than particle physics (even particle physicists themselves will, if pressed, grudgingly admit as much), and that particle physics is in turn more fundamental than condensed-matter physics, which is more fundamental than chemistry, which is more fundamental than biology, which is more fundamental than psychology, anthropology, and so on, which still are more fundamental than grubby engineering fields like, say, computer science … but then you find out that computer science actually has as strong a claim as math to be the substrate beneath physics, that in a certain sense computer science is math, and that until you understand what kinds of machines the laws of physics do and don’t allow, you haven’t really understood the laws themselves … and the whole hierarchy of fundamental-ness gets twisted into a circle and revealed as the bad nerd joke that it always was.

That was a longer sentence than I intended.

Note (Jan. 25): From now on, all comments asking what I think of the movie “Teeth” will be instantly deleted. I’m sick of the general topic, and regret having ever brought it up. Thank you for your understanding.

118 Responses to “Science: the toroidal pyramid”

1. michael vassar Says:

2. Dave Bacon Says:

“We all know, of course, that math and logic are more fundamental than particle physics ”

Doh. Lost me at step 1.

BTW, its turtles all the way down. (With quantum ones somewhere along the way.)

3. Scott Says:

Sure — and so did economics. After all, if you don’t know Bayesian probability and decision theory, then at least according to the Overcoming Bias folks you couldn’t possibly know what to believe about anything, including particle physics.

4. Isabel Lugo Says:

All sciences besides mathematics are stamp collecting.

No, I don’t seriously believe this. And some parts of mathematics might be called stamp collecting as well, including large swathes of combinatorics, which is what I do sometimes.

5. Chiral Says:

I was thinking about the fabled “Theory-Of-Everything” tee-shirt the other day. I’d like a tee-shirt that says:

Theory Of Everything:

(K x y) -> x
(S x y z) -> (x z (y z))

Universe = stamp collecting exercise left to physicists

Of course, it has no more content than “the universe is computable”, but if you’re talking about nerd jokes …

6. Tim Says:

Since combinatorics is stamp collecting, and therefore not the most fundamental science, I should procrastinate by posting a comment to Shtetl-Optimized instead of working on my combinatorics homework.

I think it was Feynman (or Gell-Mann) who said that physics is to mathematics as sex is to masturbation, so clearly mathematics can’t be the fundamental science.

Clearly neuroscience is the most fundamental science, because we do everything, including science, with our brains.

Clearly x is the most fundamental science, where x is what I work on. After all, the only reason I work on it is because it’s the most important, and if it’s the most important it must be the most fundamental.

Clearly the philosophy of fundamental science is the most fundamental science, since we clearly don’t know what the most fundamental science is, and working on anything else is a sub-optimal use of resources.

Exercise for the reader: Expand the argument above to show that 99.99% of scientists are irrational.

7. Marcus Says:

Yeah, this whole “x is more fundamental than y” thing is pretty retarded when you stop to realize that there is no unifying theory anyway. Until there is, everything is more fundamental than everything else, quite reasonably. Or maybe you just haven’t done enough acid ;-D

8. Robin Kothari Says:

“We all know, of course, that math and logic are more fundamental than particle physics”

Although the statement seems obvious to me, I have often found it difficult to convince people that this statement is true. Is there a good way to convince people about this?

Math is Physic’s language, so Math is to Physics like English is to Shakespeare’s Hamlet.

10. Nick Ernst Says:

Hmmm. Well, while I have my disagreements with David Deutsch, I’m rather fond of his take on reductionism: “…scientific knowledge consists of explanations, and the structure of scientific explanations does not reflect the reductionist hierarchy.” (larger excerpt linked here).

11. Peter de Blanc Says:

I think it may be a mistake to say that chemistry is more fundamental than biology. At least, I’m sure it is a mistake to say that chemistry is more fundamental than evolution; chemistry is a medium which can support evolution, but there are other media which can as well.

12. Jonathan Vos Post Says:

The assumptions of Reductionism are more deeply flawed than the anomalies given. For one thing, as Philsophers of Science have argued, there’s the implicit but unjustified assumption that there is ONLY ONE REAL WORLD. Is one assumes that (usually implicitly in on’e Kuhnian paradigm) then one falls into the trap of kludging together a metric for the “distance” between one’s pet discipline at that putative unique real world.

The unjustified metaphysical assumption is usually made that there is, in total, ONLY ONE SCIENCE (meaning set of all science disciplines) that describes the putative unique real world.

I’ve discussed at length with Geoffrey Landis the possibility (which would seem to make some fun Science Fiction) that we meet the extraterrestrials, and, even after years of effort, cannot understand their science or technology, nor they ours, as both of us have something that works, yet is intrinsically irreconcilable.

By the way, in teaching a history of scientific revolution course to several hundred adult students over the years, I first gave a quiz, and then again at the end, which plots metaphysical stance. Returning the first quiz, I point out that before Kuhn, essentially all scientists would have answered “true” to each of the 10 questions, and Kuhn himself answered “no.” Indeed, the quizzes were never all “yes” nor all “no” in this class, and tended to skew more Kuhnian after I presented Kuhn and Lakatos and engaged in a week of conversation with the class.

13. Bram Cohen Says:

Scott, here’s a somewhat unrelated question. If you were to build a time machine and go back and explain to Fermat Karatsuba’s multiplication algorithm and how to do a Diffie-Hellman key exchange, what do you think he would have made of it?

Perhaps another nice idea my professor in solid state physics pointed out:

For non-relativistic energies, the Hamiltonian of N electrons and M nuclei (as point charges) is a Theory of (almost) Everything since it completely determines the wavefunction of the system, which in turn completely determines all expectation values.

So from a mathematical point of view, the exercise of actually solving that Hamiltonian is left to the devoted reader.

Dirac, however, pointed out that there are two issues with that view:

a) The WF of 50 electrons in space representation is a 150 dimensional function. If one attempts to sample that function on a grid with but one bit of information per gridpoint and just 10 points in each “direction”, this would mean a space amount of 10^150 bits. I guess that is more than there are atoms in the universe…

b) By containing ALL of the information of our system, the WF contains too much information so in the end it gives to us no information at all. In particular, such a treatment does not satisfyingly explain where superconductivity comes from, why there are conductors, semiconductors and it does not allow for general observations.

So the reductionists approach does not work here. Of course, all behaviour of the solid state system relies on the underlying Hamiltonian, but this observations does not much to help us understanding the solid state system.

15. Scott Says:

“We all know, of course, that math and logic are more fundamental than particle physics”

Although the statement seems obvious to me, I have often found it difficult to convince people that this statement is true. Is there a good way to convince people about this?

I don’t think so — especially since (as my post was trying to explain) such statements actually have no clear meaning!

I do get annoyed (in fact, angry) when people claim that if physics were different then math and logic would also be. And then I try to explain why that claim makes no sense, how it confuses knowledge of mathematical truths with the truths themselves and is ultimately self-undermining (“if logic only works in our local patch of the universe, why should I listen to your logic about the other patches?”), and they respond, “oh, you must be a Platonist.” I don’t even really know what Platonism means, but I’m ready to cast my lot with it, just to stick it to the people who treat that word like “racist” or “child-molester.”

16. Scott Says:

Math is Physic’s language, so Math is to Physics like English is to Shakespeare’s Hamlet.

Or perhaps one should say: math is to good physics like English is to Hamlet, and is to bad physics like English is to a C1ali$spam. But actually I’m not happy with this metaphor, since where do discoveries in math fit into it? Are they like discoveries in linguistics? 17. Scott Says: If you were to build a time machine and go back and explain to Fermat Karatsuba’s multiplication algorithm and how to do a Diffie-Hellman key exchange, what do you think he would have made of it? Bram, that’s a tough one; as I ponder it, it seems the crucial issue is how much time I’d have with Fermat. If I had only five minutes he’d probably write me off as a crackpot (assuming I’m not allowed to show him my laptop or other such artifacts, nor the time machine itself). If I had a few hours I could completely blow his mind. (Assuming I spoke French, which I don’t.) 18. mick Says: “BTW, its turtles all the way down. (With quantum ones somewhere along the way.)” – Well it’s one turtle with 4 giant elephants on its back anyway. 19. oz Says: “I think it was Feynman (or Gell-Mann) who said that physics is to mathematics as sex is to masturbation, so clearly mathematics can’t be the fundamental science.” I highly disagree – we all know which is more fun(damental), 20. John Preskill Says: Sorry — I am reposting because my comment got a bit garbled for some reason. Is (M=mathematics) > (P=physics), where “>” means “more fundamental than”? I’m inclined to answer “of course” but it is tricky to define the question precisely. Perhaps x > y means that we can change y without changing x, but not the other way around. Then unless we believe that there is a unique consistent physical theory, we can easily imagine changing the Hamiltonian of the world without changing the laws of logic, so it seems M > P. Even if we think “1+1=2″ is a statement about physical objects, if it is a statement about physical objects in all possible models of physics, then it is really a mathematical statement, isn’t it? On the other hand, if there is a “unique physical theory” doesn’t that mean that (once we accept some suitable axiom) the form of theory follows from logic alone, so that at best M = P (an overstatement, perhaps, since we could still change P by adopting different axioms), but not P > M ? Is there a sensible way to frame the question so that the a plausible answer would be P > M? 21. Kurt Says: I do get annoyed (in fact, angry) when people claim that if physics were different then math and logic would also be. Maybe I’m not understanding what you’re saying here, but it seems to me that if the physics of our universe were radically different, then our logic and math would be too. For example, if in our world the disjunctive syllogism failed to hold for macroscopic phenomenon, our logic would certainly be different. You might counter that while our “standard” logic might be different, what we currently consider standard logic would still exist as a non-standard branch of logic in this other world. And to some extent I would agree with that. However, our brains have evolved to be able to deal with the universe in which we live, and I’m sure that there are limits to what we can comprehend in principle. If our universe were different, those limits would be different, and the intersection with what can comprehend in this world might be rather small. 22. Bobby Says: To Robin (@#8): One of the issues with communicating with “normal” people about the fundamental nature of math, is when you say math, they think you mean manipulating numbers. Most people have no idea of math as the study of all predictable systems. In fact, I’ve never used that way to portray it – maybe that could work, if you can convince people to accept your definition. I’ve tried talking about math as the study of all typographical manipulation systems, but that shoots past them too… 23. Isabel Lugo Says: To Bobby (@#22): I think that the claim that mathematics is the study of all typographical manipulation systems is true but misleading. It’s true in that all mathematical arguments (as far as I know) can be expressed as derivations in a suitable typographical manipulation system, etc. But it’s misleading in that it gives people the idea that mathematicians think about their work in terms of pushing symbols around. 24. Scott Says: Kurt, what would it mean for the disjunctive syllogism to fail for macroscopic phenomena (or for that matter, microscopic phenomena)? The disjunctive syllogism says that (P or Q) and (not P) imply Q, which is as true in quantum mechanics as in anything else. (The claim that quantum mechanics “changes the laws of logic” is so confused I barely know where to start with it.) 25. Blake Stacey Says: When I first heard about the “hierarchy of sciences” (physics -> chemistry -> biology, or whatever), back in the seventh grade or thereabouts, the question immediately jumped into my mind, “Well, where do you put mathematics, then?” Feynman’s Character of Physical Law, which I found a few years later, also left me with the image of the “hierarchy” closing in upon itself to form an Olisbian strange loop. Speaking of bad nerd jokes, I’m sure that other people here can come up with better riffs than I did for the new James Bond movie, Quantum of Solace. 26. Moshe Says: Oh, this argument about what is “fundamental” sounds like something from the 1970s. Chad has a bi-weekly post about that issue, but he is usually very thin on examples, it is usually some subtlety in wording of a sentence somewhere, or some anonymous comment on the web. Maybe I am blind to this, but is there a recent example of anyone serious and prominent that claims HEP or cosmology is more important than other fields of physics? 27. Koray Says: I am very likely to be wrong here, but I’ve always thought that the reason that we humans trace our mathematical reasoning down to the same logical axioms is because our brains share the same physical structure. Then the mathematics that we do is not independent of the physics of the universe; it could even be bounded by just the physics of our own brains. We could be like blind DFA’s that develop a grand theory of acoustics with the sort of calculations they can perform, agreeing with each other along the way that this is all there is, not being able to study the Turing machines that they cannot see. 28. John Sidles Says: I recently wrote an essay (to myself) called The Spooky Mysteries of Classical Physics … is there anyone else who considers the ontology of classical physics to be just as spooky and ambiguous as that of quantum physics? I did a literature search and found no similar essay. 🙂 29. DrRainicorn Says: To Koray (#26) Of course the way we talk about mathematics is affected by the specifics of our brains, but thats not fundamentally what mathematics is. Even if we were to use different axioms, the axioms we use now would still be true, for example, the disjunctive syllogism is true not because it makes sense to our brains, but because the way the operators it consists of are defined force it to be true. That is, if someone in another universe were to encounter a situation that encodes those operators, the truth of the disjunctive syllogism could influence events, even if they don’t discover it. And yes, the computing power afforded to us by physics may determine the amount of math we are able to discover (seemingly nothing that requires a halting oracle, for example). However the math we do discover still applies in any universe that encodes the mathematical objects we have studied (which seems very likely, given the apparent generality of such things as numbers and Turing Machines). 30. Scott Says: Koray, yes, you’re conflating two issues. It’s possible that superintelligent aliens would ask mathematical questions far beyond anything we could comprehend. But to whatever extent we asked the same questions, we’d necessarily agree on the answers. 31. Cynthia Says: Scott, slighly on, slightly off topic… Since you’re a man of complexity, I’d like to hear your answer to this question: Because Godel’s incompleteness theorems more or less affirm that math (and thus science) won’t ever be able to get a grip on infinity, do you think this adds credence to the faith-based thinkers who ascribe to the notion that faith/religion will always rule supreme over reason/science? Even if you can’t provide me with a complete answer, I’ll be happy to receive an incomplete one! ;~) 32. matt Says: Yes, I definitely feel that the ontology of classical physics is just as bad, if not worse, as that of quantum physics. When asked “why do we perceive the world as classical when it is actually quantum?” the logical answer is “why should we perceive a world as classical if it is actually classical?”. Both questions deal with the question of us “perceiving” the world in certain ways, namely a question of our conscious awareness, which classical mechanics says nothing about. You can ask the question “why do we, and other animals, model the world as classical when it is actually quantum?” and I think the answer is the same as “if you’re designing a bridge, why do you model the forces in it as if classical elasticity theory was right, when it’s actually quantum?” Namely, it’s the easiest way to do it. 33. Hatem Abdelghani Says: Dr. Aaronson, I have two points to mention regarding your post 1- The mathematics which is more fundamental than particle physics is meant to be pure mathematics. While computer science is applied mathematics. So they don’t form a circle. 2- Is it true to define math and logic as science at all? I don’t think so. They are more of philosophy and less of observations. Perhaps they form the junction between philosophy and science. 34. Scott Says: Cynthia, I don’t even really buy the presupposition of your question. The incompleteness theorems do indeed put limits on the power of any given formal system — but ironically, the undecidable sentences you get from those theorems have nothing to do with infinity! (You might be thinking about the independence of the continuum hypothesis, which is something different.) But let’s ignore that. The mistake you make is an ancient one: namely, to assume that if science can’t answer a given question, then religion automatically has the “upper hand.”. Why should we assume that? If religion is to be our guide in infinite set theory, shouldn’t theologians first have to answer the questions that mathematicians can’t? 35. John Sidles Says: Matt says: Yes, I definitely feel that the ontology of classical physics is just as bad, if not worse, as that of quantum physics … why do we, and other animals, model the world as classical when it is actually quantum? Heck, Matt, we can say something stronger than that! Let’s take Scott’s toroidal principle seriously. As I read it, the toroidal principle predicts that if we think about it, we will discover that classical mechanics is just as ontologically spooky and ambiguous as quantum mechanics. And the prediction turns out to be true: it is easy to construct “spooky” classical ontologies that link seamlessly to “spooky” quantum ontologies (two diagrams here). The idea here is not to make quantum ontology less spooky, but instead to specify a (nonstandard, but consistent) classical ontology that is more spooky. And this exercise turns out to be a lot of fun! Classical spookiness makes perfect sense, when you think about it. Given the central importance of the spookiness of quantum mechanics, it would be astonishing indeed if this spookiness were wholly invisible at the classical level! 🙂 36. Cynthia Says: Scott, my bad! ;~( Roughly speaking, Cantor’s continuum hypothesis is to infinite sets as Godel’s incompleteness theorems are to formal systems… Anyways, the *only* reason I ask you this question is because I’ve heard IDers argue the following: Since Godel’s incompleteness theorems prove that mathematical-based science will forever be plagued by either incompleteness or inconsistencies, it must look to theology/faith to obtain consistencies/completeness. 37. milkshake Says: math/linguistic analogies get you only so far – a common problem of making analogies between unrelated fields. Math to me is more like a set of tools – the constructs that have been built and thoroughly checked for bugs – and guaraneed to work if you use them correctly (thats why the syntax part of the business) 38. Len Ornstein Says: Perhaps this will stimulate some responses that will lead to clarifications? Conventional languages, logic and math require users to commit to ‘belief’ in a set of fundamentals; axiomatic rules and definitions. By following the rules, a model (conjecture) can be proven either to be true or false (or undecidable, if the model, or set of axiomatics, is incomplete or poorly-specified). All commitment, explicit or implicit involves accepting axiomatic ‘rules’ without logical proof of their ‘truth’. In this quite general sense, all deductive reasoning and the “proving of absolute logical truths”, is ultimately ‘faith-based’. Hume showed us that, using axiomatically-based deductive reasoning, an inflexible definition of an object, class or process, from any extrapolation or interpolation from observation of only a sample of its parts, is UNDECIDABLE because of INCOMPLETENESS. So there’s no logical way to justify a ‘deductive level of belief’ in the truth of any facts generated by inductive reasoning. So, to some degree, we all take facts ‘on faith’. Science depends completely on both deductive and inductive reasoning. Therefore the argument that science dispenses with the need for any elements of faith or belief – supposedly in contrast to religion – is overly simplistic. Since faith and belief are defined as metaphysical, this argument, which also is equivalent to the claim of the extensive literature of the Positivists; that science is, and must remain, free of metaphysics, is hard to accept. However, there is an important difference between the discipline required in science, on the one hand, and that of religion, mathematics and classical logic, on the other: Science requires commitment to axiomatics, in order to design robust models of reality (by carefully following the rules) so that the models can be dependably communicated and understood. But science always adds the additional injunction: a degree of belief in each model can be established only with the weight of supporting, factual evidence; the more and ‘better’ the evidence, the greater the belief. This is the important distinction between science and religion – and even between science and mathematics. (And it helps to assure that a model itself remains distinguishable from the ‘reality’ it’s been designed to simulate.) External or internal (biological) worlds, are accessed through our senses – either directly, or ‘through’ intervening ‘instruments’. All the models (ideas, hypotheses, theories, laws), constructed by theoretical scientists to generate order out of such observations – preferably, but not necessarily, carefully reasoned – starting with a conjecture and often ending with a proven theorem – nonetheless all remain agnostically tentative guesses about the nature of reality. To inspire any degree of belief that a model ‘explains’ such reality, science requires separate evidence that the model, to some degree, matches some previously unobserved (empirical) aspect of those worlds. That’s to say, to receive anything more than the most tentative consideration, an apparent deductive truth about the ‘world’ must be supported by (usually ‘new’) matching, empirical, inductive ‘truths’, (largely discounting prior similar evidence, if it has contributed to the abduction and construction of the model). This is quite unlike what’s required for establishing belief in the purely deductive proof of a typical theorem in mathematics. The number of provable mathematical conjectures (consistent models; theorems) is enormous. But the fraction of those that can be matched to worldly observations is infinitesmal. Both theoretical and experimental scientists sometimes discover, and in any case, use this tiny fraction of mathematics as extremely valuable intellectual tools. But it’s misleading to consider most of mathematics as a kind of science. Most of the time, mathematics avoids logically-undecidable models, and therefore needs no empirical matching. It ‘gets away with’ “pure reason” to prove absolutely true theorems. Science, can’t! It ‘must’ tie all of its models to messy, logically-undecidable facts – and always ends up with at least some tiny, residual uncertainty. It’s possible to generate hierarchies of models and theories of science. Strictly speaking, mathematics isn’t part of such a hierarchy of science. Locations in such hierarchic classifications are based on the generality of each component model. And the confidence with which a model has been confirmed decides whether it’s to be included. The more ‘general’, the nearer to ‘the trunk’ of a hierarchical tree of science. Many mathematicians and some scientific theoreticians – typical inventors of models – are Platonists. They (want to?) believe that some of their theories are revealing absolute and fundamental truths about ‘some reality’. But the natures of deductive and inductive truth, reviewed above, and from which, none can escape, make it ‘impossible’ to establish absolute truths about ‘external reality’– and therefore, within science, that issue should be moot. The Platonic allure of THE most general scientific discipline may provide the motivation – the challenge – to discover central ‘ultimate truth’. But within which scientific discipline will we find the most general: the reductionism of particle physics? astronomy? the physical evolution of cosmology? evolutionary biology? neuroscience? or communication/information theory and computational complexity? It depends upon how you choose to design your hierarchic tree. For example, since all mental discipline and reason which gives rise to all science must begin with language, perhaps linguistics/cognitive science and its evolutionary, neurobiological core should be the trunk? What could be more fundamental 😉 The bottom line – probably what a scientist strongly believes will decide how each values a scientific model; first, observational testability and degree of confirmation? second, economy (‘simplicity’ and/or ‘beauty’– the test of Occam’s Razor)? and then generality? Socially pragmatic scientists – those particularly concerned with human survival and comfort – might also require high utility, and even place utility first, or second? 39. Tyler DiPietro Says: “Since Godel’s incompleteness theorems prove that mathematical-based science will forever be plagued by either incompleteness or inconsistencies, it must look to theology/faith to obtain consistencies/completeness.” Actually, if we’re allowed to handwave, we needn’t resort to such intricate and turgid pablum as theology. We can just posit an oracle and be done with it. 40. Tyler DiPietro Says: “Conventional languages, logic and math require users to commit to ‘belief’ in a set of fundamentals; axiomatic rules and definitions. By following the rules, a model (conjecture) can be proven either to be true or false (or undecidable, if the model, or set of axiomatics, is incomplete or poorly-specified). All commitment, explicit or implicit involves accepting axiomatic ‘rules’ without logical proof of their ‘truth’. In this quite general sense, all deductive reasoning and the “proving of absolute logical truths”, is ultimately ‘faith-based’.” The problem with this argument is that it reduces “faith” to “assumption”, and thus generates a much weaker definition of the term than we intuitively associate with those things we consider “faith-based”. IMO, it’s not a very profound statement to say that we must assume certain things in science/math (think of the Cartesian demon), so the argument doesn’t really illuminate anything that we don’t already acknowledge. 41. Job Says: What’s not Math? Anything that is not describable? And what’s not Physics? Anything that i can’t feel or think about? From a human perspective they’re both fundamental. We’ll never come across something that’s not both Physics and Math, or one and not the other – by definition. Math doesn’t have a scope, and Physics does but has one at the edge of the spectrum. Other sciences have intermediate scopes – i.e. they run from 68 to 33. So i don’t think that there is a fundamentalness circle. Physics is at the bottom, and Math is everywhere. 42. Tyler DiPietro Says: One thing I’d like to add to the above is that while theology employs an abstract and deductive system of reasoning that is superficially similar to mathematics, the real line of demarcation comes from the presuppositional nature of theology. The “axioms” of theology are contingent upon culture, e.g., a Christian theologian must accept the divine exceptionalism of Jesus of Nazareth in one form or another. In mathematics the axioms are chosen according to their power in illuminating mathematical questions. 43. Job Says: In Theology there really is only one axiom – everything else follows. God is the axiom-explaining axiom. 44. K. B. Nikto Says: Job, axioms are not explained. They are assumed without proof. To explain a proposition is to refer it to a another proposition, one which is already known. 45. cody Says: i like your explanation response there Scott, to Cynthia. a related question: does our inability to have a complete mathematical system (strong enough to generate the reals) inhibit our ability to have a complete (or maybe better called ‘universal’) physical theory? this isnt a question ive thought about before, so ill be curious what people think. personally, i view mathematics and physics as being extremely similar, with the prime difference being that in mathematics we get to define our axioms and see what sort of behavior follows, whereas in physics we are given the behavior and we are attempting to recover the axioms. unfortunately we have no way of knowing if we have ever found all the axioms, or really, any of them. but the more our physical theories can explain, and the fewer assumptions they make, the more confident we can be that we are at least approximating the ‘ultimate’ ‘rules’. what really piques my interest in computer science is that it sits somewhere between math and physics. we dont really just define our axioms as we please, we choose ones that somehow make sense (this is usually true in mathematics too, but there seems to be more room to explore the unlikely in mathematics, which is good in many ways. id love to write more on this, but the post is really too long already. 46. John Sidles Says: As a toroidal confection, I recommend the Nebula-winning novelette Tower of Babylon by Ted Chiang. Maybe other folks can name their favorite story having a “toroidal” mathematical theme? Thinking about toroidal stories suggests that it might be fun to look for toroidal theorems. An example of such a theorem concerns the Hilbert transform. This kernel$H$of the Hilbert transform is real-valued, yet it satisfies the defining relation of what Kahlerian geometers call a complex structure, namely$H^2 = -I$. The Hilbert transform theorem suggests that we can choose to traverse the classical-to-quantum transition by descent rather than ascent (to echo Chiang’s wonderful story). Namely, we impose a Hilbert-transform invariance upon classical real dynamics, mirror the resulting complex structure in the Kahlerian geometry of the state-space, and introduce complex numbers either never, or only when we feel the practical need (feh!) to impose a coordinate system on the state-space. The result is a world in which we perceive that “spooky quantum mysteries” are manifest everywhere, even in classical physics. 47. Scott Says: Hatem: The mathematics which is more fundamental than particle physics is meant to be pure mathematics. While computer science is applied mathematics. You realize, don’t you, that you’re in a den of theoretical computer scientists? 😉 48. Alex Says: I recently got this email joke from a friend “Biologists thinks they are molecular biologists, molecular biologists thinks they are chemists, chemists think they physical chemists, physical chemists think they are physicists, physicists thinks they are God, and God thinks he’s mathematician”. It now seems that God thinks he’s Scott Aaronson…. Sorry, seems like a lame joke but just too tempting given the topic of the post. 49. Scott Says: Job: In Theology there really is only one axiom – everything else follows. Except that it doesn’t. How do you get from that axiom to, say, the evil of homosexuality or of wearing mixed fabrics? 50. oz Says: How about organizing an ‘x is the most fundamental’ contest? Examples: The physics of time-travel is most fundamental since it’ll allow as to go back in time and study all other sciences. AI is most fundamental since we could build thinking machines doing science for us. 51. niel Says: Reaching back to the Shakespeare metaphor a moment: Or perhaps one should say: math is to good physics like English is to Hamlet, and is to bad physics like English is to a C1ali$ spam.

But actually I’m not happy with this metaphor, since where do discoveries in math fit into it? Are they like discoveries in linguistics?

Shakespeare again provides the best example: one can invent in English as in math. The coining of new words and metaphors, with such art that without explanation, the audience knows exactly what is meant. And centuries after the author’s passed away, no-one recalls how anyone before could do without those handy turns of phrase.

Math, like English, is but another tongue, except that it’s an artificial one, in which to make metaphors most precise.

52. Job Says:

My point was that if you make God an axiom then you can always explain things further. You can always say because “God causes it to be that way”.

Why are mixed fabrics evil? Because God causes it to be that way.

53. Job Says:

K. B. Nikto, i was saying that if you add in the axiom of God to an existing set of axioms those axioms disappear. I’m familiar with what an axiom is – i looked it up on Wikipedia before i posted.

54. Anwar Shiekh Says:

This Computer Science verses Physics; when is it going to stop? They are not adversaries.

55. K. B. Nikto Says:

By equating the evil of homosexuality with the wearing of mixed fabrics, Scott has trivialized the important distinction between a potentially deadly sexual aberration and a sartorial choice.

56. Tim Says:

IMO, it’s not a very profound statement to say that we must assume certain things in science/math (think of the Cartesian demon), so the argument doesn’t really illuminate anything that we don’t already acknowledge.

It seems like one to the kind of people who think that an important distinction between (their) religion and science is that science has to assume things and (their) religion doesn’t (or at least not anything very important).

A few lessons on basic philosophy, the history of skepticism, Descartes’ demon, and the fact that the exploration of every non-trivial domain of human knowledge requires making assumptions and is incapable of producing real certainty would be so much more useful than the useless “observation + theory + experiment = science” crap that is the closest most people ever get to the philosophy of science (or, horrors, anything like epistemology). A lot of people pick it up more or less by by osmosis (or just assume it from the beginning, the lucky bastards), but a lot of people don’t. I didn’t get it until I was eighteen, and it took a lot of work (and blew apart my worldview). Maybe I’m unusual, but I’m willing to bet that the people who think this stuff goes without saying are not as numerous as many of them think.

57. Tyler DiPietro Says:

“By equating the evil of homosexuality with the wearing of mixed fabrics, Scott has trivialized the important distinction between a potentially deadly sexual aberration and a sartorial choice.”

You’re kidding, right?

58. John Sidles Says:

Nikto posts: … by equating the evil of homosexuality with the wearing of mixed fabrics …

Gosh that’s a ugly, hateful post. Posts like that do not belong on a good-spirited forum. No, I am not interested in excuses or debate … this is just to go on record.

59. Ryan Budney Says:

The “Pyramid of Science” I’ve always thought of as a comforting fiction that gets told to people who want those kinds of things, similar to the Easter Bunny or Santa Claus. The distinctions between the various sciences are artificial to begin with, which makes The Pyramid a fiction of a higher order altogether, and seemingly impossible to pin down. While we’re working with this vague notion of “more fundamental than”, there’s compelling arguments to say that biology is far more fundamental than mathematics, computer science and particle physics since it deals with things that people readily identify as being important. Certainly an indicator of “fundamental” should be “does it affect me in any way I’d consider important?”

60. Scott Says:

Gosh that’s a ugly, hateful post. Posts like that do not belong on a good-spirited forum.

John, I completely agree (assuming Nikto was serious). Beyond that, I’m glad to learn there’s something you hate!

61. Cynthia Says:

Nikto,

Since being gay is not any deadlier than being straight, then I say you’ve got it backwards. Wearing mixed fabric is a potentially deadly sexual aberration, whereas homosexuality is merely a sartorial choice!

62. John Sidles Says:

… I’m glad to learn there’s something you hate! …

With apologies to Alexander Pope:

Abuse is a monster of so frightful mien,
As to be hated needs but to be seen;
Yet seen too oft, familiar with its face,
We first endure, then pity, then embrace.

This goes double for bad Powerpoint presentations! 🙂

63. K. B. Nikto Says:

Cynthia, your comment is facetious and may denote a paucity of information.

64. Raoul Ohio Says:

Cynthia: “~( Roughly speaking, Cantor’s continuum hypothesis is to infinite sets as Godel’s incompleteness theorems are to formal systems…”

Other than being tough issues in axiomatic mathematics, I don’t think Cantor’s continuum hypothesis has anything at all to do with Godel’s incompleteness theorems:

CCH: The ZF set theory axioms and logic do not tell us what the size of R (the real numbers) is.

GIT: No formal system can prove all of math.

Keep in mind that R is basically a construction to enable one to prove things in calculus. E.g., when learning calculus, the intermediate value theorem sounds pretty obvious. As a math major, you learn to push around Dedekind cuts and Cauchy sequences and prove stuff like the IVT. In Topology you generalize these constructions as much as you can, and prove lots of theorems about whatever.

But who cares? No one has any idea how well R maps to any concept of space or time. This is where studying modern topology is almost as bad as being a philosophy major: it encourages bright young kids to waste their time working on pointless issues when they could be discovering better approximate algorithms for NP hard problems or better numerical schemes for NS equations. Hardy had it backwards: Useful mathematics is harder than pure math.

65. Tyler DiPietro Says:

Methinks Nikto is starting to sound a bit trollish.

66. John Sidles Says:

Quotes arrayed upon a slippery continuum …

—————————————-
G. H. Hardy: “Mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean”

Peter Lax: “Pure mathematics is a branch of applied mathematics.”

Karl Friedrichs: “Applied mathematics consists in solving exact problems approximately and approximate problems exactly.”

John Bardeen: “Invention does not occur in a vacuum. Most advances are made in response to a need, so that it is necessary to have some sort of practical goal in mind while the basic research is being done; otherwise it may be of little value.”

Deak Parsons (to Robert Oppenheimer): “Ruthless, brutal people must band together to force the FM [Fat Man] components to dovetail in time and space. They must feel that they have a mandate to circumvent or crush opposition from above and below, animate or inanimate – even nuclear!”

—————————————-

67. Cynthia Says:

K.B. Nikto,

Don’t get me wrong, I like men way too much to be gay! But IMO, you’re being somewhat naive to believe that gays have a monopoly on deadliness whether it involves micro or macroorganisms…

68. Cynthia Says:

Cody and Len Ornstein,

Assuming I’m reading some of what you’re saying right, let me restate a little bit of it in more down-to-earth terms… Since physics is merely a subset of math, then just because math is bound to run into either incompleteness or inconsistencies doesn’t necessarily mean that physics (along with the hierarchy of science) will run into either of the two as well.

69. Len Ornstein Says:

Cynthia:

Physics USES, but isn’t a subset of math.

Science differs from math (and religion) by REQUIRING matching empirical confirmatory evidence, established with some significant degree of confidence, to consider that any conjecture is a candidate as a model of ‘reality’. And that candidacy can NEVER grow to absolute certainty.

Math (with Godellian caveats), requires no more than ‘pure reason’ to prove the ‘absolute truth’ of many conjectures.

That’s a BIG difference!

70. cody Says:

Cythina, i really agree with what Len just said. as an example i would say, it doesnt really mean much to talk about a universe where gravity falls off 1/r^3, because our measurements seem to rule out that behavior. in contrast, there is nothing ‘right’ or ‘wrong’ about imagining a world with finitely or infinitely many points when studying geometry. they lead to radically different ‘worlds’, but neither one need exist for us to study it.

71. John Sidles Says:

Only in three dimensions do harmonic gravitational potentials yield periodic astronomical orbits. This is sometimes cited as an anthropic reason that our universe is 3D.

Because no other universes support the billions of astronomical orbits needed for the highest form of life … mathematicians … to evolve. 🙂

That is what the Life-universe professors deduce, anyway!

72. Stas Says:

Somewhat off topic, but how fundamental would be this science? 😉
Scott, it looks like the first author is from your lab…

73. Deja vu Says:

The disjunctive syllogism says that (P or Q) and (not P) imply Q, which is as true in quantum mechanics as in anything else. (The claim that quantum mechanics “changes the laws of logic” is so confused I barely know where to start with it.

Let’s for a moment assume we lived in a universe in which nuclear fision was so easy to achieve that every time you put together two objects you actually got only 1.5 times as much mass, with the rest being dissipated in all manner of heat and radiation. Then we would have discovered first and studied for far longer the system of mathematics in which 1+1=1.5.

Sure, the axioms stating that 1+1=2 would still hold, just like the axioms stating 3+5=2 mod 6 still hold. However their role in mathematics would be less central.

Or to use a more real example, non-euclidean geometries were a curiosity within math for quite a while, until we discovered that we might actually live in one such after which their role in math become more important.

74. Scott Says:

Let’s for a moment assume we lived in a universe in which nuclear fision was so easy to achieve that every time you put together two objects you actually got only 1.5 times as much mass … Then we would have discovered first and studied for far longer the system of mathematics in which 1+1=1.5.

Deja vu, that sounds cool if you don’t think about it too much, but I think it completely collapses if you do. What exactly is “the system of mathematics in which 1+1=1.5”? Your use of those words doesn’t conjure such a system into existence (much less a unique consistent one).

Here’s one way of explaining the difficulty: even to state the rule that “every time you put together two objects you actually get only 1.5 times as much mass,” you needed the ordinary concepts of 2 and 1.5, didn’t you?

75. Scott Says:

Stas: Thanks for the link! I feel that my colleagues’ work richly deserves, and will quite possibly receive, an Ig Nobel Prize.

76. Hatem Abdelghani Says:

You realize, don’t you, that you’re in a den of theoretical computer scientists?
Sure, but that doesn’t make the cycle closed yet. Theoretical computer science, which lies at the beginning of the chain of fundamentalism, is not the same as applied computer science, which lies at the end of that chain and which belongs to engineering more than it belongs to mathematics.

That’s my opinion anyway.

77. Cynthia Says:

Cody and Len Ornstein,

At risk of falling into the black hole of semantics, let me extract this tidbit from both of your comments…

Math is a tool for both science and religion. But while science uses math to grasp reality, religion uses it to create fantasy.

78. Deja vu Says:

I think it completely collapses if you do. What exactly is “the system of mathematics in which 1+1=1.5″?

Easy “plus” instead of describing the natural (there’s that pesky word again) addition, defines a completely formal function f on two variables that has the property that f(x,x)=1.5 x, f(x,0)=x,
f(x,y)=f(y,x), etc.

Your use of those words doesn’t conjure such a system into existence (much less a unique consistent one).

I just did, and it is fully consistent. I can now study the theorems and properties behind the very useful function f. Is it associative? Is it continuous?

Here’s one way of explaining the difficulty: even to state the rule that “every time you put together two objects you actually get only 1.5 times as much mass,” you needed the ordinary concepts of 2 and 1.5, didn’t you?

Gee, that means nothing: say, if I’m an eskimo and I need to describe a camel for the first time I will use concepts that are familiar to the eskimo (say, it’s like a bear, but longer legs and has a hump like a whale). If on the other hand I’m an australian aborigene I’ll use concepts and words familiar to those who live in the outback.

So you see, the use of a given set of words does not prove their centrality or universality, but simply our familiarity with them.

Don’t get me wrong: the deep ideas of math are universal, but the specific systems we study, the shorthands we use to describe them and the order in which we study them are very much a reflection of our physical reality.

79. Scott Says:

Deja vu: You haven’t fully defined the function f. Can you please do so? What is f(2,1), for example?

80. 12Quarts Says:

What basis is there for believing that all of science has developed anything but a slightly more “fundamental” understanding than was achieved by 12th century theology?
Sure, it seems like the technology developed off the science is huge, but how much more is out there that we aren’t aware of? How do you know but that ev’ry bird that cuts the airy way is an immense world of delight, closed to your senses five?

Maybe mathematics and logic as we know it is only a subset of the sort of thinking that really gets things done–derivatives of a truly fundamental way of approaching knowledge of our universe. Maybe physics has been leading us down only a marginally useful path for the last 300 years.
Maybe…

81. John Sidles Says:

12Quarts Says: What basis is there for believing that all of science has developed anything but a slightly more “fundamental” understanding than was achieved by 12th century theology?

That is a serious question, and so quoting comedic lines from old Blackadder episodes is IMHO not the most appropriate answer.

I would say that a major advance is humanity’s still-growing and increasingly multidimensional understanding of our own natural history. This theme unites cosmology, biology, cognitive science, and evolution … it unites every branch of science and mathematics.

Humanity’s understanding of our own history and literature is deepening too … and this includes our understanding of Scripture. To read the books of Moses and *then* read Frans de Waal’s Chimpanzee Politics, is to participate in an adventure that was possible to no previous generation.

As with any adventure, a considerable degree of effort, discomfort, and inconvenience is entailed, and of course the destination of the journey is uncertain too (this uncertainty is a major theme of Donald Knuth’s writing on this topic).

Obviously too, our understanding is a work in progress … we are still wandering in the desert, and no one knows when (or even if) we will reach the Promised Land! 🙂

82. Raoul Ohio Says:

Since I was a kid I have toyed with the question of how much of our math and science will turn out to be the same as that of alien civilizations when we establish radio communication. What is fundamental and what is arbitrary?

Assuming enough math to support radio engineering, here is my model:

1. Counting is pretty basic to intelligence. I will guess a 99% likelihood of the same N we have.

2. Adding and subtracting are also basic. 95% for the same Z.

3. How do they think about division? Many reasonable systems such as an extension of ieee 754 FP numbers would work fine. 90% for the same Q

4. I think our reals are a lot more arbitrary. Do you even need to consider the solution to x^2 – 2 = 0 to be a number? 50% for roughly the same (Dedekind cuts, Cauchy sequences) R.

5. If you are working with radio, you probably have discovered something like complex numbers. 75% they make the same extension as we do to whatever they use for R.

If anyone wants to post their estimates, I offer a \$1 bet about who comes closer with whatever civilization contacts us first. It might take a while to find out who wins. Those going to Heaven (surely including St. Don) can keep tabs for us.

Finally, what are the technical concepts you would have to stumble onto as you develop technology? Fourier transforms, Hilbert space; Heap sort, Quick sort; Quantum (mechanics, computing, info th)? Anyone wishing to study this might assess independent rediscovery as a measure of how obvious a concept is. Shell sort seems to be kind of an outlier. Does anyone know if it has been frequently rediscovered?

83. cody Says:

Cynthia, i dont see math as a tool for religion at all. i would say that mathematics is the language in which science is ‘spoken’. though math on its own is more than just a language or tool. and yes, id agree that science uses math to ‘grasp reality’. but ‘religion’ in general does not use math at all; to the contrary, it is heavily reliant on tradition and fantastic lies to explain reality (poorly).

84. Joseph Hertzlinger Says:

When I was an undergraduate, I came up with a “proof” that all science was circular based on the following:

Psychology is really biology.
Biology is really chemistry.
Chemistry is really physics.
Physics is really mathematics.
Mathematics is really logic.
Logic is really philosophy.
Philosophy is really psychology.
Psychology is …

85. John Sidles Says:

There is also a toroidal aspect to the cardinal tenets of the Enlightenment, as conceived by mathematicians and scientists in the 1600s and 1700s:Reason as the sole criterion of what is true.Rejection of the supernatural.The equality of mankind.Ethics stressing equity, justice, and charity.Comprehensive toleration and freedom of thought.Personal liberty.Freedom of political criticism and the press.Democratic republicanism as the most legitimate form of politics.(this particular list is from Israel’s Enlightenment Contested.)

These tenets were conceived by the authors of the Enlightenmen—who included the most distinguished mathematicians and scientists of those centuries—as uniquely elevating humanity “above” the animal kingdom. But as we learned more about our own evolutionary history, we have come to see these tenets not as moral verities founded upon mathematical logic (Leibniz’ view), but rather as a contingent mirror of our own evolutionary history.

In other words, the moral principles of the Enlightenment, being devised by primates, are satisfactory to primates, but would be less satisfactory to societies descended from cows, and wholly unsatisfactory to societies descended from ants.

Ed Wilson famously made this point when, upon viewing the movie Aliens, he remarked “Whoever made this movie knew a lot about social insect morality!” And we can add, primate morality too … the whole movie can be viewed as an extended morality play.

So in seeking to escape our primate heritage, we toroidally come to understand and embrace it more thoroughly.

Even today, many folks (including me) seek via mathematics to embrace simplicity, morality, and truth … The good news is, we get to “pick any two.” 🙂

86. Cynthia Says:

I totally agree with you, Cody! Theologians aren’t mathematical heavyweights by any stretch of the imagination. But many of them are masters of numerology and other such cryptic nonsense.

87. Deja vu Says:

Deja vu: You haven’t fully defined the function f. Can you please do so? What is f(2,1), for example?

Our imaginary reality would give you the definition. Just like our current reality tells you that bodies in free fall do so with a constant acceleration (you can imagine a universe in which bodies fall at say, a quadratic acceleration, can you?)

88. John Sidles Says:

To defend Deja vu’s point of view, IMHO he is absolutely right to intuit that nonstandard classical models of reality exist that are both mathematically consistent and perfectly useful as practical models of reality.

A well-known example is the Feynman-Wheeler (FW) classical electrodynamics, in which (classical) electrodynamic propagators are half-advanced and half-retarded.

The FW picture of reality agrees with experiment, and is mathematically consistent too … it simply is not the way that we are conventionally are taught that classical reality “works.”

For example, FW physics has the “spooky” property we are all subject to “classically real” electric and magnetic fields that impinge upon us … from the future!

We are taught that electric and magnetic fields are “physically real”, but as FW physics explicitly demonstrates, this seemingly clear definition of classical reality has some mighty spooky aspects. And when we study noisy classical systems, then the notion of “simple classical reality” becomes even spookier.

It is characteristic of Feynman’s way of thinking that he noticed—and took seriously—the notion that classical reality is similarly spooky to quantum reality.

The good news for students is, classical reality is still mighty spooky today, fifty years after Feynman began wondering about it. We really have not made much progress in understanding this.

89. Scott Says:

Yes, there are useful nonstandard models, but the point is that Deja vu can’t even give me a model! He’s not going to tell me what f(2,1), f(3,2), etc. are, and for good reason: as soon as he fully defined his nonexistent universe, I would catch him in an absurdity.

It’s like this: if I’m having a strange nightmare, all I need to do to wake up is ask one pointed question: “No, why are you telling me this, if before you were telling me that?” The nightmare-generating part of my brain can’t compute an answer to the question, and the entire nightmare crashes. I’m trying to use the same tactic to wake up from Deja vu’s nightmare where 1+1=1.5. The only problem is that he’s not answering the questions.

90. John Sidles Says:

Scott, to respect both you and Deja vu, it is surely true (as you point out) that most nonstandard models of reality simply don’t make sense, being inconsistent and/or incompletely defined and/or at odds with experience, so why waste time with them? Life is too short!

Yet Deja vu is right too (IMHO), to point out that the number of models of reality that *do* make sense is strictly larger than one, even at the classical level.

On the rare occasions that new models can be found that work, they are fertile sources of new mathematics and physics. So much so, that the discovery of a new model is a pretty good working definition of what constitutes “new mathematics and physics”.

Example: Turing machines as models of computation and cognition. No one claims that cognition actually works this way … but what a great new model!

The 21st century too will find new models … and IMHO the most novel aspect of these 21st century models is likely to be their informatic and algorithmic content.

If only we knew what these new models are going to be … 🙂

91. harrison Says:

Even though there are plenty of nonstandard models out there, I think they’re pretty boring. Noncommutative rings/fields (coughquaternionscough) are really the only interesting “different arithmetics” I’ve ever seen. A lot of what mathematics is is saying “if you want a model that satisfies this and this and this axiom, then you’re stuck with something isomorphic to this.” So a system of addition in which 1+1 = 1.5 (whatever that even means) wouldn’t follow the “normal” rules of addition.

Also: C’mon, people, no one’s made the obvious Ricoh “non-standard model” joke? This is a travesty.

92. Job Says:

Joseph Hertzlinger, i don’t agree with your jump from Logic to Philosophy. I find that arguments for the circularity of sciences tend to use a little bit of word play (derived from a certain ambiguity) and are more poetic than rational.

For example, to argue that Chemistry is more-fundamental-than Biology you seem to be using the fact that Chemistry is-used-by Biology. But when you go from Logic to Philosophy you seem to be using the makes-use-of relation- because i don’t see how Philosophy is-used-by Logic. I think that you are jumping from Logic to Philosophy because the first is a subset of the second – but that jump is not in keeping with the previous jumps – or at least i don’t see how it is.

Psychology makes-use-of biology.
Biology makes-use-of chemistry.
Chemistry makes-use-of physics.
Physics makes-use-of mathematics.
Mathematics makes-use-of logic.
Logic makes-use-of philosophy.
Logic is-a-subset-of philosophy.
Not coherent
Philosophy makes-use-of psychology.
Psychology makes-use-of …

In addition, Philosophy makes use of everything and Math is used by anything – so i would leave these two out because they don’t add anything meaningful and just lead to confusion and circular arguments.

93. John Sidles Says:

Harrison says: Even though there are plenty of nonstandard models out there, I think they’re pretty boring.

Isn’t that simply because a really successful nonstandard model pretty swiftly becomes the standard model?

Rapid progress in biology in recent years supplies plenty of examples. For example, a closed gene pool has become the standard definition of what constitutes a species.

However, it is far less clear (to me) that this same process works in mathematics & perhaps it does not work at all.

It is downright disheartening, for example, to read in the introduction to Joe Harris’ Algebraic Geometry, that a textbook so long that (in Harris’ words) “only someone who was truly gluttonous, masochistic, or compulsive would read every example” is merely an introduction to a more advanced mathematical ontology that focuses upon sheaf cohomology and scheme theory as expressing the “real” essence of algebraic geometry.

Yikes! There’s not much empirical evidence of toroidal cognitive topology in algebraic geometry! Instead, the mathematics of algebraic geometry (and perhaps many other branches of mathematics too) is apparently becoming endlessly richer, without becoming simpler.

94. Greg Egan Says:

In the interests of finding out precisely where Scott vs. Deja vu is going, can I propose a candidate for f?

f(x,y) = 3(x+y) / (3+xy)

This has f(1,1)=1.5, is commutative and associative, but not distributive over ordinary multiplication. It has the usual identity, 0, and the usual additive inverse, f(x,-x)=0. It’s not defined when xy=-3, though one remedy for that would be to assume a universe where the relevant physical quantities must always be less than sqrt(3) in magnitude, a property that f will preserve.

I thought Deja vu’s point was that it’s conceivable as a matter of alternative physics and culture that a society might arise that considers f to be a more “elementary” or “intuitive” operation than our addition, and would construct their axioms and definitions accordingly. But maybe I’m misunderstanding the debate, because that possibility seems uncontroversial to me.

95. Scott Says:

Greg, thanks for suggesting an explicit model for f! (An interesting one too.) Your model reminds me of the velocity addition rule in special relativity (with √3 as a “speed of light”) — something that might also be proposed as “more basic than addition” for some alternate civilization.

But on further reflection, I wish directly to controvert the possibility that seems uncontroversial to you! That is, I wish to propose not only that addition will have the usual properties for every civilization, but also that addition will seen as more basic than f or the SR addition rule by every civilization.

My argument, as usual, rests on the impossibility of even talking about the other operations in a coherent way without already knowing what addition is.

There are many ways to flesh out this argument; here’s one of them: can you give an algorithm for computing 3(x+y)/(3+xy), which does not invoke ordinary addition or anything trivially similar to it as a subroutine? (Since the notion of an algorithm is conceptually prior to that of addition, I don’t think there’s any circularity in this question.)

96. Scott Says:

Incidentally, in giving an algorithm to compute 3(x+y)/(3+xy) that doesn’t use addition, no fair exponentiating and then multiplying and taking logs! 🙂 A civilization that has those operations already has a structure that’s trivially isomorphic to addition.

One other remark: as an unexpected corollary of my “algorithmic argument,” I can imagine a civilization that would discover the integers mod 2 before it discovered the integers, or the bitwise-AND and bitwise-XOR before it discovered addition. It’s just functions like 3(x+y)/(3+xy) that I’m having severe difficulties with.

97. Sumwan Says:

Well, if you consider the trigonometric identity
tan(a+b) = (tan a + tan b)/(1-tan a tan b) (I hope it is correct) , perhaps that there is a civilization that considers that the operation of summing 2 numbers is just getting the tangent of the sum of their arctangents. One can find flaws in this argument, e.g that you need to have geometry to define tangents, that to have geometry you need lines, to have lines you need real numbers, to have real numbers you need rational numbers (at least in the constructions I know of), to have rationals you need natural numbers and to have natural numbers you need the successor operation, which is the same as adding one.
But intuitively , it seems possible to me that an outer space civilization having access to some kind of analog computer in their bodies view this operation, or one similar to it, as an elementary one.

98. Deja vu Says:

I thought Deja vu’s point was that it’s conceivable as a matter of alternative physics and culture that a society might arise that considers f to be a more “elementary” or “intuitive” operation than our addition, and would construct their axioms and definitions accordingly. But maybe I’m misunderstanding the debate, because that possibility seems uncontroversial to me.

Exactly. Even more so if we assume that such culture arose in a hypothetical universe where the laws of physics are different.

I can imagine a civilization that would discover the integers mod 2 before it discovered the integers, or the bitwise-AND and bitwise-XOR before it discovered addition.

This is exactly my point. They would then describe addition in terms of modulo 2 operations and the fundamental axioms would be given in these terms. If we had a universe in which things added up like the f function I suggested, sooner or later they would discover our addition (as well as mod operations) but the description would be in the other direction: ‘+’ is just like f but instead of giving 1.5x gives 2x !!

In fact for all we know the truly central concept in mathematics might be, say, complex numbers, and we are left having to describe them in terms of square roots of negative numbers (square roots being a notion that we discovered through geometric areas) simply because our grasp of electricity is not as built in as our grasp of spatial geometry. If we were eels, perhaps we would have discovered i first and then describe the naturals as an odd subcase of complex numbers in which the imaginary part is 0.

99. John Sidles Says:

With respect, maybe you guys aren’t posing hard enough math questions?

Suppose the aliens send us a multiple-choice mathematical test (the same one they send to thousands of civilizations).

But the aliens grade harshly: if we answer wrong, then they send out the Berserker-type von Neumann machines.

100. Scott Says:

Just like our hypothetical aliens, the Sumerians, Babylonians, and ancient Chinese knew of many cases where “addition” did behave in nonstandard ways. For example, they knew that distances at right angles add as √(x2+y2), and that the fraction taken if you take an x fraction and then a y fraction of what remains adds as x+y-xy. Yet without exception (so far as I know), it was ordinary addition, and not these other operations, that they took as fundamental. Why?

Of course, the burden of answering this question doesn’t lie with me; it lies with those who think math could as easily be based on the other operations as on addition. Personally, I think the answer has almost nothing to do with physics or the human brain, and almost everything to do with the structure of mathematical ideas.

What I’m proposing here is a falsifiable hypothesis: you could falsify by building a mathematical theory that makes as much internal sense as the usual one, but that takes √(x2+y2) or x+y-xy or 3(x+y)/(3+xy) as a basic operation and x+y as a complicated derived one. But I can’t stress that enough that you’d actually have to build this theory: it’s not enough to claim that you can imagine it being built.

101. cody Says:

is anyone else here a strict physicalist, tending to think that the whole of the universe, and all its associated phenomena are (potentially) explainable with a set of (quite possibly unobtainable) physical laws? because that has always been my motivation for the hierarchy of the sciences (physics before chemistry before biology before psychology).

in my mind, computer science is almost independent of reality somehow, but more concrete than plain mathematics.

102. Kurt Says:

Scott said:

Kurt, what would it mean for the disjunctive syllogism to fail for macroscopic phenomena (or for that matter, microscopic phenomena)?

I don’t know. That’s the whole point. Now, the disjunctive syllogism was just the first logical inference rule that popped into my mind; there may be more interesting examples that could be used. But since it can be thought of as kind of an operational definition of the word “or”, a universe without the disjunctive syllogism would be a universe without the concept of “or-ness”. Could such a thing exist? I don’t know, and again, that’s the whole point.

Deja vu is getting beat up a bit in the comments, because his/her example is not nearly radical enough. How about this: Our human language is structured around a distinction between noun and verb, actor and action. Our brains have evolved this capacity (presumably) because the universe is this way: clumps of matter coalesce together into discernible entities which can interact with each in various proscribed ways. What if our universe did not have at its base matter, energy, space and time, but something totally different? I cannot conceive of what that might be like, because I am of this universe. However, I am not willing to therefore conclude that it couldn’t be otherwise. That is what I mean when I suggest that our logic and mathematics is a result of our physics.

(By the way, my inclusion of the word “macroscopic” was not meant to imply anything about quantum mechanics, but only to make explicit that I wasn’t trying to say anything about quantum mechanics. I seem to have set off some kind of woo-sensor of yours.)

103. Scott Says:

…I wasn’t trying to say anything about quantum mechanics. I seem to have set off some kind of woo-sensor of yours.

Yes, sorry — when you work on foundational issues in quantum mechanics, you have to check your woo-sensor like a Chernobyl cleanup crew checks its Geiger counter.

While I won’t say that the world you describe can’t exist, there’s nothing whatsoever we can say about it, precisely because it’s outside all of our linguistic categories by definition. So we might as well go back to discussing something else! 🙂

104. Job Says:

If we were to assume that our Universe can be simulated with a minimum of two distinct data states and two distinct operations (whatever that is) with an arbitrary amount of space/time, then a given Universe would only be different from ours if it either did not support the required minimum states/operations or if it supported any state or operation not derivable from the two fundamental states/operations. Right?

105. Greg Egan Says:

Scott (#95): yes, I stole the SR velocity addition rule for f, though I didn’t want to highlight that because I didn’t want to drag along all the physical baggage of velocity addition. And of course Sumwan’s example (#97) is very similar, because it’s just the Euclidean version of the Lorentzian velocity addition rule.

But my choice of f was a bad one for motivating exotic mathematics, because it’s isomorphic to ordinary addition, i.e. ((-√3, √3),f) is isomorphic as a group to (R,+), with G(z)=√3 tanh(z) as the map from (R,+).

In a world where objects of weight x and y tended to meld together into composites with weight f(x,y), I suspect that the notation and language would just reflect the origins of objects, rather than their final weights, and hence it would amount to working in (R,+). Normally, they’d simply talk about “two standard-weight glubs having been melded together”, as a way of describing f(1,1), rather than comparing the melded weight to an unmelded sum and needing to compute 3(x+y)/(3+xy). In other words (translating their notation, whatever it is, into our binary), their “101” would mean a weight of f(4,1), and their “110” would mean a weight of f(4,2), but they would still add these numbers by a process isomorphic to our binary addition, and get “1011”. Most of the time, f(…) would just be an implied “wrapper” around the notation; it wouldn’t need to be explicitly evaluated.

Of course, when they did want to make the comparison between melded weights and the arcane concept of a sum of unmelded weights, they might indeed compute something which, in their notation, amounted to expressing x+y in terms of f (and I think a second operation would be needed as well). But for this example, at least, I have to come down on Scott’s side: I think this culture would just work with the variables that made things simple, and end up doing ordinary addition.

106. Alex Says:

Scott, why do you think the idea that quantum mechanics changes the laws of logic is confused? Isn’t that the whole idea behind the filed of quantum logic (as in Orthomodular lattices and non distributive logics not the “quantum logic” of C-not and Hadamard circuits). Here’s a quote form the Stanford encyclopedia of philosophy:
“At its core, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic.”

107. Scott Says:

Alex: Yes, I’m familiar with quantum logic, and I can explain why many of the claims made for it are hogwash.

The fundamental problem is that the outcome of a measurement you could’ve performed but didn’t is not an event. So if you plug it into logical expressions and manipulate it as if it were an event, you shouldn’t be the slightest bit surprised if you run into apparent paradoxes.

By analogy, suppose a squirrel is known to be hiding under one of two bushes, and suppose you look under one of the bushes and don’t find it there — but suppose the very fact of your looking disturbs the squirrel, causing it to run away from the other bush (where it was previously hiding), so that you don’t find it under the other bush either. Have you therefore disproved the Disjunctive Syllogism, that (P or Q) and (not P) imply Q?

The question itself is so ridiculous, it sounds like the setup for a Marx Brothers skit — yet from my point of view, many of the philosophical claims made on behalf of quantum logic are just as willfully dumb.

None of this is to attack the mathematical study of the lattice of subspaces of Hilbert space, which some people enjoy (I’ve never been one of them). Just please don’t confuse people by calling the subspaces “propositions”!

108. Scott Says:

Greg: Thanks for pointing out what should’ve been obvious to me right away, that your 3(x+y)/(3+xy) system contains a subsystem isomorphic to (R,+)! In retrospect, that was of course inevitable, as soon as you defined a commutative, associative operation f:R2→R with identity and with basic continuity properties. (Exercise for readers: which assumptions are actually necessary?)

109. Alex Says:

Thanks for the nice illustration. Why have there been so many papers published on the subject (just wondering)?

110. Deja vu Says:

Yet without exception (so far as I know), it was ordinary addition, and not these other operations, that they took as fundamental. Why?

Because most people add objects first and foremost. Children at a very early age (2 year olds) discover on their own the concept of addition.

What I’m proposing here is a falsifiable hypothesis: you could falsify by building a mathematical theory that makes as much internal sense as the usual one, but that takes √(x2+y2) or x+y-xy or 3(x+y)/(3+xy) as a basic operation and x+y as a complicated derived one.

That has already been done. Many times over. Set theory for one makes x+y a complicated operation. What are otherwise natural and simple concepts of integer and addition become a nested mess of sets of the empty set (4:= {{{{{}}}}}).

Here’s another one: non-euclidean geometry makes as much sense as the classical geometry, yet basic concepts such as lines, geodesics, parallelism and angles behave in different ways.

Here’s one more: you can define logic in terms of the basic AND and OR operators as classical mathematicians did or you can build it around NAND and XOR as computer scientists did. AND now becomes a “complicated derived operation”.

Of course, the burden of answering this question doesn’t lie with me; it lies with those who think math could as easily be based on the other operations as on addition.

Contrary to what you say the burden of proof is on you. You are making an outlandish claim: the uniqueness of mathematical axioms.

I on the contrary believe that failing formal proof it is much easier to imagine alternative consistent models and in fact have given you examples of such.

In fact, it takes no effort to see that this is the case. Let us imagine for a moment that the Earth had a much smaller radius. Would have classical geometry where parallel lines do not intersect been discovered first? of course not. The geometry of the sphere would have been studied first with its many interesting and yes, self-consistent properties.

111. Scott Says:

Deja vu, your approach is similar to that of someone who claims that cheese graters can grant eternal life, and then, when the evidence starts stacking up against that hypothesis, indignantly exclaims, “but science doesn’t know everything!” No, but that wasn’t the question.

Of course addition can be built out of the successor function, or out of Boolean AND and XOR operations. In an earlier post, I said myself that I could imagine a civilization that discovered Boolean logic prior to addition.

But we weren’t talking about that. Nor were we talking about whether non-Euclidean geometry could have been discovered before Euclidean geometry, which is an interesting but separate question. We were talking, specifically, about your hypothetical system of arithmetic where 1+1=1.5.

Let me repeat: we were not talking about the uniqueness of axiom systems in general (of course they’re not unique, in general). We were talking about whether your “weird addition” could have been discovered before ordinary addition.

To fill you in, we’ve made some actual progress on this question. After you repeatedly refused to give me a concrete model for your nonstandard addition function, Greg Egan was kind enough to do so. However, Greg then realized that while his function has many nice properties, precisely because of those properties it’s isomorphic to ordinary addition (the mapping being the hyperbolic tangent function). (Earlier, I had pointed out that the hypothetical civilization presumably couldn’t compute his function without using ordinary addition as a subroutine.)

I’m now interested in the question of exactly what nice properties an addition function can have without being isomorphic to the ordinary one, and/or without requiring ordinary addition to compute.

So that’s where the discussion is at. If you’re able to contribute to it, please do so. If you keep raising irrelevant strawmen, I’ll have no choice but to block you for trolling.

112. Joe Shipman Says:

You can have a “nonstandard model” (countably infinite) of addition and multiplication which satisfies all the same sentences as the ordinary integers under the usual + and *, and you can make either operation computable, and the possible resulting submodels for addition-only or multiplication-only are well-understood.

What you can’t do is have BOTH the nonstandard addition and the nonstandard multiplication be computable. So it’s hard to point to an explicit “nonstandard model”.

113. John Sidles Says:

Recognizing that nonstandard models of reality are hard to construct, Wheeler and Feynman deserve credit for a pretty good try in their 1949 article Classical Electrodynamics in Terms of Direct Interparticle Action.

Even though their nonstandard model of (classical) reality was not as successful as they hoped, the article is still a nice case study in how to go about constructing such models.

One lesson-learned is that such enterprises require a *huge* effort and a *lot* of calculation.

Also, if space aliens tried to confuse us by transmitting an electrodynamics textbook written in this nonstandard idiom, they would probably succeed! 🙂

114. Greg Egan Says:

Given that every Lie group has subgroups isomorphic either to (R,+) or U(1) — and I’d describe addition on U(1) as so close to that on R as to be morally equivalent — it’s hard to imagine a culture where the vital concept was a Lie group not considering (R,+) or U(1) even more fundamental.

But suppose the most important physics and culture revolved around, say, the Klein 4-group. That’s {0,a,b,c}, with a commutative addition such that x+x=0 for all x, and a+b=c and all permutations thereof.

Sure, Klein addition is simple, but it seems to me conceptually independent of conventional integer addition; you could argue that x+x=0 is integer addition modulo 2, but I’d say that’s a degenerate case more primitive than the general concept of addition.

115. John Sidles Says:

Greg sya: Given that every Lie group has subgroups isomorphic either to (R,+) or U(1) …

Greg, I’m no expert, but if memory serves, isn’t the addition of points on elliptic curves also associated with an (Abelian) Lie subgroup? That embedding Lie group being the (heh! heh!) toroidal Lie group associated with the (doubly periodic) Weierstrass elliptic function?

So let’s imagine an alien civilization that is wholly focussed upon acquiring one scarce resource …. WiFi bandwidth. And within that civilization, the one way to acquire that bandwidth is … break each other’s public key exchange algorithms.

The resulting civilization would experience rapid biological evolution, via the well-known “Peacock’s Tail” mechanism: “Hey baby, check out my high-bandwidth big-screen internet connection! It looks so good cuz I’m tapping into every WiFi transceiver in the neighborhood!” 🙂

Over the millenia, their biological brains would become hard-wired (in the Chomsky sense) to conceive of of the addition of points on elliptic curves as being the most “natural” and “obvious” element of mathematics.

We might find their alien mathematics to be mighty hard to decipher. Indeed, their initial communication to us might looks very much like a string of random numbers. The idea being (from their alien point of view) that we humans have to prove our sapience by recognizing and responding to it as the first half of a public key exchange algorithm.

Cuz duh, that’s obviously the first thing that civilized galactic races do … exchange public keys! There being no other logically possible basis, for inter-species trust and cooperation!

Indeed, supposing that this race knows a whole lot more about information theory than we humans do, such that they bases their civilization’s key exchange protocols not upon elliptic curves, but upon “obviously better” algorithms that we humans just haven’t conceived (as yet), then it might be mighty tough for us humans to recognize their initial communication as anything other than noise.

That’s pretty much how I feel about modern algebraic geometers, anyway. Folks like Alexander Grothendieck are obviously from some other planet! 🙂

116. John Sidles Says:

Another thread bites the dust … here’s a summary. 🙂

117. KWRegan Says:

I’ve actually opined that Grothendieck’s work may hold a key to progress on P vs. NP, though along that line, Ketan Mumuley’s joint work at least starts on our terra firma.

One can flatten Scott’s pyramid into the “Penrose Triangle”—see “On Math, Matter and Mind”, which addresses the circularity issue. My own interpretation is that if the reductionist hypotheses that give rise to Joseph Hertzlinger’s chain (comments 84, 92 here) are taken all-together, then one gets a stack whose bottom is not a turtle or elephant, but rather “universe-is-a-computer” in Seth Lloyd’s sense.

118. Jonathan Vos Post Says:

“… Over the millenia, their biological brains would become hard-wired (in the Chomsky sense) to conceive of of the addition of points on elliptic curves as being the most “natural” and “obvious” element of mathematics….”

And this lobe of their brains hosts a highly optimized factorization algorithm for large semiprimes.

Vernor Vinge (after he retired as full-time Math prof at San Diego State U.) to devote himself to being the great full-time science fiction author that he is told me (I think at last year’s Westercon) that he’s been imagining aliens with transfinite computational brains, for whom, if you ask them ANY question in integer arithmetic (i.e. Diophantine) find the answer instantly obvious.

“Ummm, about those Godel numbers…?” I said.

He just grinned.