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.

]]>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.

]]>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! 🙂

]]>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.

]]>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! 🙂

]]>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”.

]]>*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.

]]>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.

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