[Basic] Skill of the week: Mathematics

[Ulzgoroth]

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?

Maybe in the future it'll change, but I'm quite sure modern mathematicians very often aren't using computers for any significant part of their work.

Sure, Mathematics/TL8 includes computer-assisted proof methods, but they're not a major distinguishing aspect.

[whswhs]

Originally Posted by Ulzgoroth

Maybe in the future it'll change, but I'm quite sure modern mathematicians very often aren't using computers for any significant part of their work.

Sure, Mathematics/TL8 includes computer-assisted proof methods, but they're not a major distinguishing aspect.

Of course, proofs are mainly important in Mathematics (Pure). For Mathematics (Applied), you care more about computations and estimates. Numerical methods were a big part of that as far back as Newton, but computers have boosted the effectiveness of numerical analysis quite a lot.

Likewise Mathematics (Statistics). Back when principal components analysis and factor analysis were first invented (partly as an outgrowth of IQ testing), if you wanted to extract the first factor of the variables for a dataset, you gave a graduate student a lot of paper and a mechanical calculator and asked them to come back at the end of the summer. Now a computer can get as many factors as you like in seconds, either in order of variance explained, or rotated to yield equal weights.

[Ulzgoroth]

Originally Posted by whswhs

Of course, proofs are mainly important in Mathematics (Pure). For Mathematics (Applied), you care more about computations and estimates. Numerical methods were a big part of that as far back as Newton, but computers have boosted the effectiveness of numerical analysis quite a lot.

Likewise Mathematics (Statistics). Back when principal components analysis and factor analysis were first invented (partly as an outgrowth of IQ testing), if you wanted to extract the first factor of the variables for a dataset, you gave a graduate student a lot of paper and a mechanical calculator and asked them to come back at the end of the summer. Now a computer can get as many factors as you like in seconds, either in order of variance explained, or rotated to yield equal weights.

Pure, Computer Science, and Cryptology. But you're right, Applied, Statistics, and Surveying tie in considerably more with tool use.

...Tangentially, I'm curious about the borders between Cartography and Mathematics (Surveying)...

[Polydamas]

Originally Posted by Mithlas

Or the adventure couldn't happen, in which case a skill roll can only hamper the GM.

No, in that case they have to give up that one plan, and think of another (climbing the wall at the mouth of the valley, smuggling themself in in a shipment of supplies, being appointed ambassador, ...) or they find themselves at the bottom of a rope 50' from the bottom of the valley and get to have some MORTAL PERIL.

Originally Posted by Mithlas

I think that's a very demeaning view of human science and history.

Its the view I got from actual computer scientists when I was studying that: people working in the mathematical end are glad to admit that they can't follow every detail of some modern proofs, and that a reasonable number of published proofs contain errors because they are pushing at the edge of what the human brain can follow (which is another reason why some mathematicians rely on intuitive, pattern-recognition "does that smell right?" instead of following the argument step by step like a computer). A diagnostic that a mathematician is working in the former mode is the phrase "it is intuitively obvious ..." which my parents' mathematics professors were reciting in the middle of the 20th century and which one can probably find earlier.

[Ulzgoroth]

Originally Posted by Mithlas

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.

Modern technology and organization to a large extent operates on a basis of making sure that no one human ever needs to juggle that much at once. You split things out so that each person has their own work that they can understand, and they don't need to understand everybody else's in detail.

[Phil Masters]

Originally Posted by whswhs

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).

How much does any of that relate to, derive from, or feed into the recognisable technologies of those TLs? Could many or any of those changes plausibly have come sooner or waited until later, without serious consequences for the physical technologies in use?

(Honest, non-rhetorical question. I hit my personal Maths Brick Wall at around 17.)

[whswhs]

Originally Posted by Phil Masters

How much does any of that relate to, derive from, or feed into the recognisable technologies of those TLs? Could many or any of those changes plausibly have come sooner or waited until later, without serious consequences for the physical technologies in use?

That's a hard question for me. A lot of TL7-8 math is at too high a level of abstraction for me; I completely failed to grasp the logic of Galois's arguments when I got them in abstract algebra, for example.

I think there are actually two subquestions here. First, how much of science can be done without math; second, how much of technology could have been arrived at empirically, without science.

On the second, it seems to me that TL6 is the point where the science starts to come first. At TL5, the invention of the steam engine came first and was what led to the development of thermodynamics (starting with Sadi Carnot's work on heat engine efficiency). But at TL6, you have organic chemistry leading to the aniline dye industry, and Maxwell's electromagnetic equations leading to radio. And that becomes even more important at TL7-8.

For physics, at least, meaningful work without mathematics is impossible after TL5. Thermodynamics, statistical mechanics, electromagnetism, relativity, quantum mechanics, and physical optics are all math from one end to the other (though not ONLY math; I learned the hard way that knowing the math isn't sufficient if you don't have the physical intuition to guide you in applying it_. That's less true in chemistry and still less in biology, though I don't think we could have gotten molecular genetics without X-ray crystallography, which is an intensely mathematical technique—and without understanding the structure of DNA, a lot of the innovative technologies of the last half century would have been stillborn.

On the other hand, there are fields where the application of mathematical techniques may have been less productive or even done harm. Philip Mirowski's More Heat than Light is a really interesting study of the problems of mathematical economics, for example.

[whswhs]

Originally Posted by Polydamas

Its the view I got from actual computer scientists when I was studying that: people working in the mathematical end are glad to admit that they can't follow every detail of some modern proofs, and that a reasonable number of published proofs contain errors because they are pushing at the edge of what the human brain can follow (which is another reason why some mathematicians rely on intuitive, pattern-recognition "does that smell right?" instead of following the argument step by step like a computer). A diagnostic that a mathematician is working in the former mode is the phrase "it is intuitively obvious ..." which my parents' mathematics professors were reciting in the middle of the 20th century and which one can probably find earlier.

You remind me of my favorite joke about mathematics:

A professor is giving a lecture on mathematical logic. He writes an expression on the blackboard [this is an old joke!], and says, "Then it is obvious that. . . ."

After a pause, he says, "Excuse me, please."

He walks to another blackboard at the side of the room. There he writes a series of expressions, muttering to himself, and occasionally crossing one out. This takes about fifteen minutes.

After that, he comes back to the front of the room, and says, "I was right! It IS obvious that. . . ." and carries on.

[whswhs]

By the way, let me note that this discussion exemplifies what I love about the GURPS community. . . .

[johndallman]

Originally Posted by whswhs

By the way, let me note that this discussion exemplifies what I love about the GURPS community. . . .

Definitely.

To add a few more things on the relationship between mathematics and technology, the simple concept of a computing machine that Turing devised, for the pure mathematics purpose of trying to discover what can and can't be calculated, was immensely influential in the design of actual machines, because it told would-be inventors that a particular design would be able to do useful work.

Something else that was very influential was Shannon's master's thesis, which showed that Boolean logic, which had existed for decades, could be used to simplify the design of electronic circuits that calculated logical and mathematical functions. This mean that there was a sound basis for engineering them, rather than just making them up. Eleven years later, Shannon also created a sound theory for communication, which identified the limits of what's possible.

Those results from the 1930s and 1940s still form the basis of modern computing and communications.