[Basic] Skill of the week: Mathematics

[whswhs]

Originally Posted by Polydamas

I like the biological and philosophical examples.

I suppose that a counter-example to "if we can envision it, we can do it" would be Fermat's Theorem: imagined as possible in the 17th century, first published proof which most peers accept in the late 20th.

Perhaps P=NP will one day be a comparable case.

There's a kind of ambiguity here. Predicting the orbits of the planets was something that could be imagined very early; the ancient Babylonians and the Maya were doing it at TL1. But they used horribly clunky approximative methods that couldn't be translated into physical models in any way that made sense. Then Kepler came up with the theory of elliptical orbits with the sun at one focus. At that point it was possible to envision finding exact solutions to astronomical problems—but that insight was, in essence, the answer to the problem. It wasn't a matter of "being able to envision the result" but "being able to envision the method." (Though ironically, Kepler's method relied on conic sections, which had been worked out at TL2. No one visualized them as solving the problem of planetary orbits, though.)

[Phil Masters]

My Hermetic Cabalist character, who's from an early 18th century timeline, has managed to end up with Mathematics/TL4 (Surveying) because of his old day job, /TL2 (Pure) because he's a Verdadera Destreza stylist, and /TL6 (Applied) because he's been studying the physical sciences in some more advanced timelines.

He doesn't roll against any of them much, and anyway to him they're all means to an end; he's a philosopher (and a would-be eikone, but that's a long-term project).

[malloyd]

Originally Posted by whswhs

Then Kepler came up with the theory of elliptical orbits with the sun at one focus. At that point it was possible to envision finding exact solutions to astronomical problems—but that insight was, in essence, the answer to the problem. It wasn't a matter of "being able to envision the result" but "being able to envision the method."

Though you can look at it as being a massive simplification instead - epicycles do after all more or less work to predict planetary positions - which makes this look more like a mathematical tech advance for which you can find similar cases. Negative numbers simplifying accounting problems, or complex numbers or Fourier transforms turning problems involving confusingly time variable stuff into simpler ones that don't, or reciprocal lattices making diffraction patterns calculable. Perhaps there are analogous transformation waiting to be discovered that will turn n-body central force problems or incredibly messy fluid flow ones that require vast amounts of number crunching to approximate into simpler ones yielding exact solutions.

[johndallman]

Originally Posted by Polydamas

since a few decades ago, a significant number of proofs rely on computers to check them. No human can follow all hundreds of pages of close reasoning without error. I think that may have been involved in the Fermat proof.

Some proofs rely on this, the first being of the four-colour theorem in 1976. It's not especially common, because there are mathematicians who object to the idea.

It does not appear that computer proof was involved in Andrew Wiles' proof of Fermat's Last Theorem. He didn't mention the idea in the lecture I saw about it in 1995, and according to the Wikipedia page about his proof, a challenge was set in 2005 to formalise a proof that can be verified by computer, which kind of suggests it didn't exist at the time.

[whswhs]

Originally Posted by malloyd

Though you can look at it as being a massive simplification instead - epicycles do after all more or less work to predict planetary positions - which makes this look more like a mathematical tech advance for which you can find similar cases. Negative numbers simplifying accounting problems, or complex numbers or Fourier transforms turning problems involving confusingly time variable stuff into simpler ones that don't, or reciprocal lattices making diffraction patterns calculable. Perhaps there are analogous transformation waiting to be discovered that will turn n-body central force problems or incredibly messy fluid flow ones that require vast amounts of number crunching to approximate into simpler ones yielding exact solutions.

It seems to me that such transformations would indeed be new methods that we haven't yet envisioned.

[Humabout]

Something I'd be curious about is where the lines between TL 5, 6, 7, and 8 mathematics get drawn. While it may be difficult to predict what happens at future TLs, we can finish doing for math what LTC1 started.

[Mithlas]

Originally Posted by Polydamas

Using basic surveying trigonometry to figure out how much rope you will need to descend at night into the secret valley, as in a Gordon Dickson novel, is a good adventuring use of Mathematics (Suveying).

Or the adventure couldn't happen, in which case a skill roll can only hamper the GM. I think you're only reinforcing my statement that this is a skill with applications better served by other skills. I don't have my books for a list right now, but Accounting or Law or Architecture would be a better skill to roll to spot a conspiracy's money trail or the critical spots on a powerplant.

Originally Posted by Humabout

I would also allow it as a complimentary roll for a WIDE variety of skills - Engineering, Physics, Chemistry, Cryptography, Navigation, Accounting, Finance, Economics, Administration, Politics, Marketing, Merchant, Pharmacy, and Physician just off the top of my head.
...or if using RPM, it could stand in for Thaumatology pretty easily.

I think this could make sense.

Originally Posted by Polydamas

I like the biological and philosophical examples.

I'm told that already in the middle of the 20th century, many mathematicians relied on intuitive feel for whether a proof was valid

I've read several 'history of 0 or some other math concept', and rigor is something that we've applied for centuries. For example, the vitriol resulting from Newton and Leibniz' came from a lot of people looking very closely at the mechanisms. "Because it feels valid" isn't even something that's considered acceptable in philosophy where internal consistency can be more important than real-world relatability.

Originally Posted by Polydamas

No human can follow all hundreds of pages of close reasoning without error.

I think that's a very demeaning view of human science and history. Since Charles Babbage proposed what we would understand as a (modern) computer in 1871, it took only to 1931 for both infrastructure and human science for Alan Turing to start the mechanical calculator to defeat the German Enigma. Or the limited tools and information available for astronomy still allowing viewers centuries before Galileo to propose the idea - mathematically - that some of the 'irregularly moving stars in the sky' were planets all moving with earth when satellites and precise telescopes weren't available.

[whswhs]

Originally Posted by Humabout

Something I'd be curious about is where the lines between TL 5, 6, 7, and 8 mathematics get drawn. While it may be difficult to predict what happens at future TLs, we can finish doing for math what LTC1 started.

Well, that's not all that difficult. Between 5 and 6, you acquire non-Euclidean geometries and the concept of curved space; you acquire multidimensional methods such as quaternions, vectors, matrices, and tensors; you see the arithmetization of analysis, where instead of talking about infinitesimals and ratios and sums of infinitesimals, you talk about limits; you see the emergence of the idea of a system of postulates as defining a mathematical entity or system (for example, the Peano postulates for the natural numbers); you have the proof that algebraic equations of fifth or higher degree have no general solution, and the emergence of group theory from it. Later at 6, a lot of that math moves into physics and gives us relativity and quantum theory.

The big thing between 6 and 7 is the discovery that formal logic has limits in mathematics, by way of the Gödel-Church theorems. The Turing proof about the halting problem is parallel, and in the course of working it out Turing comes up with the concept of automata theory, which gives birth to Mathematics (Computer Science). You also see the full efflorescence of fields such as abstract algebra, which will feed back into elementary particle physics with the identification of symmetry groups—an example of Mathematics (Pure) turning into Mathematics (Applied).

I think the big things in going to TL8 mathematics mostly emerge from the presence of computers. Of course there are efforts to prove theorems by computer. But more important, I think, is the recognition that some systems are not predictable, even if we have exact equations for them, because they have sensitive dependence on initial conditions, and an error below the limits of precision of your computations can blow up to change the outcome in a big way (the famous "butterfly effect"). You also have the emergence of computability theory and the measurement of computational complexity. And Mathematics (Cryptography) takes off with the recognition of number theory as having practical significance.

I'm sure a professional mathematician could point out other important developments that I've missed. But I think most of these are fairly big ones.

[Peter Knutsen]

Originally Posted by whswhs

Well, you know, the fundamental contradiction is that if we could envision TL12 mathematics then we would HAVE TL12 mathematics. It isn't as if you needed to create new machines or synthesize new substances or anything to do the math. Math is designed to run on organic brains, and we have those brains.

Unless, I guess, you want to imagine improved organic brains with higher math ability than ours. Perhaps at TL12 every kid is born with Intuitive Mathematician.

Doesn't higher TL versions of skills assume the availability of tools? Including tools such as electronic computers for Mathematics at our TL, and possibly at least primitive AIs for Mathematics at TL10 or 11?

[whswhs]

Originally Posted by Peter Knutsen

Doesn't higher TL versions of skills assume the availability of tools? Including tools such as electronic computers for Mathematics at our TL, and possibly at least primitive AIs for Mathematics at TL10 or 11?

That's a possible direction to look in, yes. See for example the fams in Kingsbury's Psychohistorical Crisis.