An Essay On The Psychology Of Invention In The Mathematical Field

Lately, there has been a lot of discussion on the nature of invention in circuit design. It turns out that the big names in mate and physics have had similar experiences and use similar approaches.

Jacques Hadamard wrote a book on this, which is available online (see the first reference in the Wiki).

Who was Hadamard? Let me put it this way: He was an insider at the Einstein level in the development of Quantum Mechanics and Relativity, and could just call up all the big names of his day on the phone.

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Reply to
Joe Gwinn
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Best known for the Hadamard transform patterns of 0 and 1 that can be used to form an orthogonal basis set also called Walsh functions.

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Used in lossless image compression and error correction on some satellite probes. I first encountered them as Walsh transforms.

His work in this field could yet become fantastically important again if large scale quantum computing ever really takes off.

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Regards, 
Martin Brown
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Martin Brown

And in IS95 CDMA cellphones. They are used to create the

64 separate channels/users available at each base station.
Reply to
Tom Gardner

Cool. Thanks. I ordered the paperback.

He does distinguish between discovery and invention. I guess there is a diminishing number of possible mathematical discoveries, and they keep getting harder to find and prove and get accepted. There's no limit on circuits that can be designed, and the possibilities keep increasing, and you don't have to convince some bunch of skeptics that something works.

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John Larkin      Highland Technology, Inc 

The best designs are necessarily accidental.
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jlarkin

George Polya wrote a number of books on what he called Heuristics, including a short extract called "How To Solve It."

Cheers

Phil Hobbs

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Dr Philip C D Hobbs 
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Phil Hobbs

Maybe- but Hadamard had absolutely nothing to do with any of that.

Reply to
Fred Bloggs

It's available as a free PDF of the 1945 edition. Just a little 145 page pamphlet. Plain talking, doesn't seem abstruse, lots of repetition, don't know why he even wrote it. Maybe you can find some inspiration there.

Reply to
Fred Bloggs

Way to offend a mathematician, mucking up their beautiful mathematics with talk of pedestrian stuff like "applications."

"Nice equation, but what is it _good_ for?"

Reply to
bitrex

Yes. I have that too. It's excellent.

Joe Gwinn

Reply to
Joe Gwinn

Having known a lot of mathematicians, that is an exaggeration in the same vein as claiming that electronic engineers fume when people ask why they need 8.5 digit voltmeters.

But you know that.

Reply to
Tom Gardner

Reply to
bitrex

I studied computer science at art school, which is where the dilettante children of the wealthy and privileged go to learn to throw a streak of paint on a canvas and call it "art" while the classes are entirely about indoctrinating 18 year olds to believe that 2 + 2 = 5 and there is no material difference between men marrying women, women marrying women, women marrying dolphins, or men marrying an 807 radio valve.

This is true and false in equal measure depending upon what subset of the "art school community" one tends to associate with, except for the part about throwing a streak of Krylon or one's old urine or bat guano on a canvas and calling it "art." This is extremely legitimate art to my mind, not in scare quotes. And it also tends to annoy the bejesus out of authoritarians of all types and American conservatives particularly, which is an excellent value for the money.

I suggest a starting bid for the piece "Untitled, matte acrylic & poop on canvas" of $7000.

Reply to
bitrex

On a sunny day (Tue, 15 Dec 2020 14:34:36 -0500) it happened bitrex wrote in :

These days neural nets Maybe one day neural nets will do better math and ...find better mathematical ?ways? I say after all we are neural nets, math is just a model in a small sub-circuit of neurons, that has been shown with MRI.

I am not denying math, use it all the time (more or less) and the code systems it found are cool (see GPS for example).

But the application of it is what counts.

For solutions.. to catch the ball.. the mama-ticion needs so many parameters, the computah crunches away .. time passes by, as does the ball. And the small neural net in the head of the sea lion makes it catch the ball with - and balance it on its nose. So much for relativity :-)

Each day one more field seems taken by AI

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It is HARD to describe the AI reasoning from math POV beyond the smallest nets.

Reply to
Jan Panteltje

On a sunny day (Tue, 15 Dec 2020 21:43:50 -0500) it happened bitrex wrote in :

+1
Reply to
Jan Panteltje

On a sunny day (Tue, 15 Dec 2020 10:15:57 -0800 (PST)) it happened Fred Bloggs wrote in :

I say the wheel was invented first, PI came later often mama-ticians claim the reverse for many things. I do not really blame them, after all they went in their early school days through the horror of having to divide by zero! This seems to have caused lasting damage for some; "re-normalization" was a cure?

Reply to
Jan Panteltje

They can be used as tools to do grunt work that humans would find way too tedious to enumerate all the possibilities. The four colour map theorem was the first serious problem to fall to computer aided proof in

1976. That is quite along time ago.

Neural nets may well allow computers to see patterns in mathematics that we as humans cannot (or at cannot until someone discovers and documents it and then it may well be obvious and taught at school). eg calculus

+1

but we are rather complex NNs. It will be a while before computer simulations of AI surpass a human brain in terms of number of nodes and uinterconnects. But they may still have a win if they can simulate that part of the brain that deals with abstract thought and reasoning.

Relativity is one of the most fully tested and verified physical theories on the planet. Errors in major computer programs have been found by looking at deviations from the predictions of the programs and observations of distant pulsars in close proximity to Jupiter or Saturn.

You cannot ask a neural network why or how it has come to a conclusion - that is a big disadvantage but if it finds the right answer you can check that in other ways. Sometimes knowing where to look is half the battle and computers don't give up so easily as humans.

You don't understand it - that is a different thing entirely.

I am not sure why but this seems to be endemic with electronics engineers. I presume it stems from the bad teaching of it in university engineering courses. Either you accept Maxwell's equations and from that those axioms that relativity is right or you are on a hiding to nothing.

Relativity was inevitable from the moment that Maxwell derived the wave equation for electromagnetic radiation, but it took Einstein to unify the whole thing into a coherent self consistent world view.

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Regards, 
Martin Brown
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Martin Brown

That was a problem with the four-colour proof, and even with human generated proofs where only a small number of people worked in the field.

Of course that won't prevent corporations and politicians using such tools to assess people and even jail them (see comp.risks for examples).

Reply to
Tom Gardner

On a sunny day (Wed, 16 Dec 2020 09:08:47 +0000) it happened Martin Brown wrote in :

Sorry Martin, that was meant as a joke.

As to relativity, Einstein's has little meaning to me without a MECHANISM.

He was basically a 'field theory' person, there is HIS description of reality. Alain Aspect's

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Einstein, in his last days, acknowledged he failed to unite gravity with the other forces. And we still do not have a mechanism,

This all is incomplete, just this morning I was reading about the farthest away galaxy and how they measured the redshift:

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it is IMO very dangerous to use an incomplete model (in this case the propagation of EM waves) to make claims. Are all those constants (speed of light etc) fixed since the bang or whatever it was? I do not want to go into endless threads like sci.physics once had, but some, what's the word? CAUTION would be in place to accept ANY modern quantum(for example) crap (it largely is in my opinion) as reality. Here back to our neural nets, go back just a few thousand years, earth wind and fire were the basic elements if I remember my history lessons well.... Would be cool to read about our _current_ models of reality a few thousand years from now. What would be left of it. I have one experiment I want to try (after tritium decay that I stopped now) related to 'dark matter', There was an other article a while ago that attributes the effect of the orbits of outer stars in galaxies not to 'dark matter' but to the pull of other neighboring galaxies:
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We are still learning As to math in all this, applied to it, we observe something, then use math to make a model so we can predict things say Volta found I = U / R, and then find that for R is infinite there was still current flowing in a vacuum tube. and then all of the sudden had a mechanism with electrons quantized at that. So I take what we have now at the very very most as an incomplete or even completely wrong description of reality. CAN we have proof? Sure we can confirm our models of reality (that is what it is) to some point by experiment. OUR experiments, in THIS time and space, with OUR equipment, We are no more than the ant creeping up a wall, it has no notion of the builder, architect, composition, probably just follows the smell of the scout ant that went before it, or smell of food. A huge amount of papers about QM has as the last line; "and this will bring the quantum computer much closer" fundraising . Or "and Einstein was so proven right again" need to get past peer review, scout ants. Same for claiming putting out less CO2 will stop warming. As to all that science; it is as corrupt as things get, Maybe wars are the way to clear it all out, we have seen in WW2 how it gave us missiles (Von Braun) Radar (UK) etc etc and in fact allowed us to leave earth and walk on the moon. In wars what works is used, UK scientists claimed V1 V2 could never work because gunpowder did not heave enough energy to carry those missiles all the way to the UK. To bad for the UK those missiles did not use gunpowder for propulsion. Better stop here...

Reply to
Jan Panteltje

The mechanism is that it makes the laws of physics the same for all observers in an inertial reference frame and with a bit of cunning algebra it can be derived from the mutual events of two metre rules passing each other at velocity v using only high school algebra.

And everybody since has failed to do this. One day we may understand why and have better grand unified theory that encompasses everything we know so far as a weak field limiting case. I am not holding my breath. One of my contemporaries is a world leading string theorist and a brilliant mathematician.

Funnily enough people have conjectured that the constants of nature might change with time but tests of that have shown that they don't change appreciably on any scale or distance that we can observe.

It is even considered (and then rejected) in Astrophysics textbooks like Harwit's Astrophysical Concepts (ISTR Dirac first made the conjecture). I remembered correctly here is his 1938 paper I think free access:

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They would almost certainly have a much more complicated mathematical model to describe more of reality but it would still have to encompass everything that relativity and QCD predicts just as relativity reduces to Galilean/Newtonian dynamics when v now) related to 'dark matter', There was an other article a while ago

Dark matter originally meant anything not emitting light under its own steam (ie non-stars). Observational techniques have no improved so much that you cannot hide dark matter as planets, sticks of rhubrarb or old biros or in fact as anything that interracts with electromagnetic waves.

It isn't completely wrong any more than Galileo was wrong. It is right in its applicable domain but you can always hope for a better theory that is able to give the right answer in the more extreme test cases.

General relativity has done pretty well on that score predicting what a black hole would look like and what a black hole merger will do to spacetime. What is truly amazing is that the experimentalists have been able to make the observations with enough signal to noise to confirm it.

One of my supervisors obituary was "Ed Shire: a device to destroy the flying bomb". He was one of the inventors of the radio proximity fuse.

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Regards, 
Martin Brown
Reply to
Martin Brown

I like real books.

I was thinking about writing something about invention myself, so I may as well review other peoples' thoughts.

There doesn't seem to be a lot of introspection in engineering or in the sciences. Scientists squabble about as much as old hens. Some engineers think they are the Pope.

--
John Larkin      Highland Technology, Inc 

The best designs are necessarily accidental.
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jlarkin

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