Math is White Privilege...

Maybe. I doubt Torvalds would have created Linux had he gone to school in Hawaii. It took an environment where nights lasted 6 months.

Other than that, I'm going to have to credit Arabs and Indians with our modern numbering system, algebra and astronomy. And all from hot climates.

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Paul Hovnanian     mailto:Paul@Hovnanian.com 
------------------------------------------------------------------ 
Steinbach's Guideline for Systems Programming 
        Never test for an error condition you don't know how to 
        handle.

On a sunny day (Sat, 28 Oct 2017 10:25:31 -0700) it happened "Paul Hovnanian P.E." wrote in :

Hot climates today, but in those days? Sahara once was full of plants. The last major glaciation was only 20,000 years ago (few hundred generations?) Those changes can go very fast, I remember scating in my youth in Amsterdam. These days ? It is much warmer now.

Last year I do hardly remember seeing any ice, just partly frozen water.

He does, but it's not clear to me how useful those conditions may be.

Godel's theorem proves that some true statements are unprovable. The simplest example is the statement "this statement is unprovable". As soon as it is proved, it becomes false; so therefore it cannot be proved. Therefore it is true, because (as just demonstrated) it's unprovable.

Cute, but useful? I don't think so. This kind of problem arises whenever you generalise a system enough to allow it to talk about itself. Russell's paradox was effectively "this statement is false". Its principle of operation is the same as Godel's proof. The Theory of Types was constructed to make such self-referential statements illegal, or at least to give them special status. Which they do! They have the status of being able to take on truth values other than "true" or "false". The two additional values are Paradoxical and Undecidable (the latter has a form like "this statement is true"). These truth values have the same role in logic as sqrt(-1) has in mathematics... but AFAIK, this has not yet been well explored.

Clifford Heath

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The argument works - more or less - for the population that lies within rou ghly one standard deviation of the mean. If you wanted to say much about th e populations in the outer tails, you'd need new tests with items that very smart or very dumb people got wrong 50% of the time.

Being able to ace an IQ tests means that you aren't doing it in the way tha t it was designed to be done, and different skills are coming under test.

There also the point that different people find different items hard and ea sy. IQ lumps together a lot of different abilities and some people put in a lot of work trying to tease out separable factors, without be able to come to any useful conclusions.

IQ tests are a quick and dirty device to roughly separate people along a si ngle poorly defined "smartness" axis. It's useful only when you haven't got the time or money to spend on doing something more informative, and worryi ng about whether what is being measured is actually "normally" distributed is paying more attention to the measurements than they deserve.

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Bill Sloman, Sydney

Not quite. It is that in order to prove other statements, some statements have to be defined as axiomatically true.

Not quite; that is the "Cretan Paradox" which is 2.5 millennia older, from the Cretan philosopher Epimenides.

Russell's paradox was "does the set of all sets contain itself"

Ahmmm... well...

Pretty much everything of note IS, essentially, all self-referral. Try defining "mass and force" independently, or "consciousness"

Also

"First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015)"

Is a statement that simply stands on its own.

Just as QM and the speed of light were additional axioms to classical mechanics, I see no basic reason why new required axioms are not going to always crop up in the future. A theory of everything is pretty much delusional in my view.

-- Kevin Aylward

- SuperSpice

That's also been known for centuries. It's called an "axiomatic basis".

Godel's work is far more subtle than that, but what I said is close enough.

Yes. but the structure of that is the same as the simpler version I gave. Russell's version is in terms of "a set" which he's already defined; "a statement" was not defined already. He had hoped that his definitions would somehow prevent the paradox, but it cannot be prevented. The stricly dichotomy between true and false is incomplete, and that undermined all the rest of his work, which is why he saw it as a catastrophe. It wasn't until he was near death and was introduced to George Spencer-Brown's work that he saw (with some relief!) that it was not actually a catastrophe. And neither is Godel's.

Clifford Heath

oughly one standard deviation of the mean. If you wanted to say much about the populations in the outer tails, you'd need new tests with items that ve ry smart or very dumb people got wrong 50% of the time.

Cite?

hat it was designed to be done, and different skills are coming under test.

Cite?

easy. IQ lumps together a lot of different abilities and some people put in a lot of work trying to tease out separable factors, without be able to co me to any useful conclusions.

single poorly defined "smartness" axis. It's useful only when you haven't g ot the time or money to spend on doing something more informative, and worr ying about whether what is being measured is actually "normally" distribute d is paying more attention to the measurements than they deserve.

Cite?

Dan

Depends on what you mean by "useful". It describes an upper limit on exactly what portions of mathematics are automatable.

But Russel was trying to derive all of mathematics from first principles

- and failed.

That's only insignificant if you don't think about it too much; mainly, if you're principal interest in mathematics is modelling only physical phenomena.

Never mind that both Turing and von Neumann both derived really significant work from that.. and that it pretty much ends the idea that "if you do all the steps right you'll get a good answer", the sort of Logical Positivism that should have died out at the end of the

19th Century.

But shoot yeah, it's useful! After all , isn't the Two Generals problem but a variation on the Halting Problem? And my career has been nothing but explaining the Two Generals problem, over and over again to people in over their heads.

Things that describe what cannot be known are much more useful than mere knowlege :)

The way I explain Godel to EEs is that a Godel-sentence does not latch to a single measurable value, but describes an oscillator. That's

*really* sloppy terminology but it's OK as a metaphor.

Which is interesting - it sort-of rhymes with your "sqrt(-1)" metaphor

- you can't, say, describe a EM sine wave in 2-space, really - requires

3-space and it's really a "spiral".

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Les Cargill

roughly one standard deviation of the mean. If you wanted to say much abou t the populations in the outer tails, you'd need new tests with items that very smart or very dumb people got wrong 50% of the time.

It's a personal opinion. I do know at least one academic expert on IQ testi ng, but I'm not all that impressed by him, or the subject. Spearman justifi ed extracting a single factor which correlated over a "gallimaufry tests" b ecause he through that there was an underlying single factor but nobody has ever found one.

that it was designed to be done, and different skills are coming under tes t.

Personal experience.

d easy. IQ lumps together a lot of different abilities and some people put in a lot of work trying to tease out separable factors, without be able to come to any useful conclusions.

a single poorly defined "smartness" axis. It's useful only when you haven't got the time or money to spend on doing something more informative, and wo rrying about whether what is being measured is actually "normally" distribu ted is paying more attention to the measurements than they deserve.

The personel guy at ITT-Creed where they'd been IQ testing everybody for ma ny years. I asked him if the results of the test correlated within they fou nd about the tested people once they'd been working for ITT-Creed for years , and his reaction was that they correlated with absolutely nothing, which tied up with what I'd read elsewhere. They are sort of useful for working o ut how many kids will do at school, and slightly less so for undergraduates . After that it's a complete waste of effort - happily, not very much effo rt.

Mensa exists to prove that you can have a high score on a high IQ test with out being distinguished in any other way. The people who do well without ev er having scored high on a IQ test don't feel any urge to advertise the low scores - they have other stuff to boast about.

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Bill Sloman, Sydney

The problem with IQ tests AISI is that you can *train* for them. And of course a good many people do - and consequently score better against those who haven't trained, all other things being equal. This renders the results essentially meaningless IMHO.

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Haha!

Hmmm. I think Russell's does that. Godel's describes a bi-stable that you can't observe (turn it on, what state is it in?). As I said before, there are two additional truth values, paradox and Undecidable. George Spencer-Brown suggests that paradox is like sqrt(-1) for logic, but doesn't realise that the same problem occurs with "this statement is true" aka a bistable, which is undecidable. Have a read of his "Laws of Form" sometime. Or email me and I can send a PDF. I'm not hard to find.

Together, the four logical states give rise to a whole arithmetic which my ex-brother-in-law has been investigating for decades. It reproduces the arithmetic of quantum numbers, which is very interesting! Extend into Noyes bit-string physics, and you can derive the cosmological constants... from pure logic (with the right "choices").

Clifford Heath.

You can also train for long distance running. The point is that "all other things" aren't equal, and the effects of training don't make enough difference to render the results of the test significantly more meaningless than they were in the first place.

The IQ test is a poor measure of an ill-defined quality, and letting people prepared for the test doesn't make the tests much more useless than they were to start with.

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Bill Sloman, Sydney

I'm sorry - there is a more-general belief in "oscillating paradoxes" and I'd attributed it poorly here.

It's quite easy to find as it turns out.

Indeed.

Wow!

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Les Cargill

On Oct 29, 2017, snipped-for-privacy@ieee.org wrote (in article):

Well, the US Army alone tested millions of people taken more-or-less at random from the general population, during WW1 and WW2, so the tails of the distribution are well explored, down to parts per million, so we can be pretty sure down to at least three sigma, which gets one down to the 99th percentile or higher. One would expect that the historic geniuses were taken from the 99th percentile (IQ above 160), not the 1th percentile (IQ around

30). So even if the tails aren?t exactly gaussian, they are close enough.

Also, given that IQ is known to be wildly polygenetic, one would expect a gaussian distribution, by the law of large numbers.

All these things have been debated for at least a century, and true once-in-a-century genius seems to transcend IQ tests. But it?s a numbers game - the higher the average IQ of a population, the greater the number of really smart people, the more things that will be discovered or invented. And the more geniuses there will be.

Beware - the Chinese are coming.

It may be quick and dirty, but it?s held up for a century as the single most useful measure - despite its many flaws, nobody has succeeded in replacing it. Not for lack of trying.

Joe Gwinn

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At the tails of the distribution, you get no questions right, or all the qu estions right, so the test tells you nothing about those individuals (beyon d the binary fact that they are very dumb or very bright). Pushing millions of people through the test makes it very likely that the test has been app lied to the tails of the distribution, but doesn't extract any more informa tion.

One always expects a Guassian distribution. Measuring what's actually going on is the interesting thing to do, and a single plain vanilla IQ test appl ied to the whole range of abilities doesn't tell you much about the extreme s.

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At least one of them showed up at Cambridge Instruments when I was there, a nd while the comapy didn't hang onto him for long, he did revolutionise the way it it together the electron microscope column. Before he arrived, each of the three pole pieces in the column had to be aligned by twiddling thre e grub screws per pole piece, and the column was easy to knock out of align ment.

His innovation was to have the pole pieces and the housing machined to unpr ecedented accuracy (which had only recently become commercially feasible), and accurately enough that you had to heat the housing and cool the pole pi eces (with liquid nitrogen) before you could slide the pole pieces into pla ce. Once they had been shrink-fitted together they stayed aligned.

The existence of smart Chinese doesn't guarantee that their society is all that wonderful at finding these people, and sending them off to get educed. Our guy had been such a brilliant machinist that his Chinese bosses had se nt him off to a Chinese university, and his Chinese University had been so impressed with him that they'd sent him off to Cambridge UK to do a Ph.D. o n electron microscopes. He didn't think much f the system that had picked h im out, and got head-hunted from us by an American firm that could offer hi m - and his wife and kids - instant American citizenship. He'd been in the UK long enough to be entitled to apply for UK citizenship by then, and he w ould have got it - eventually - but the American offer guaranteed US citize nship, back before Dubbya and Trump had devalued it.

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It's not the single most useful measure. Universities still use examination s to do much the same job, and get graduate students to write theses which are examined by panels of experts, and the results of these tedious and exp ensive procedures are taken a lot more seriously than IQ test results, whic h have the overwhelming advantage of being quick and cheap, and the equally overwhelming disadvantage of not correlating strongly with anything we nee d to know.

University entrance in Australia isn't based on IQ tests, but on the result s of written examinations.

Some 60% of the entrants to Australian Universities emerge with a degree, a nd your changes of emerging with a degree correlate relatively weakly with your exam results (once you get below the top 10% who have a 95% change of emerging with a degree). The universities are aware of this, and use other information to bias their choices towards the students with lower exam resu lts who happen to be more likely to stay the distance.

IQ tests are popular with academics who specialise on IQ tests. It's a more useful interest than working out how many angels can dance on the head of pin, but not all that much more useful.

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Bill Sloman, Sydney

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The left does understand that it's "not all nuture".

Steve Pinker published the book in 2002, and there hasn't been any kind of left-wing counter-blast.

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At the tails of the distribution, the very dumb get all the questions wrong and the very bright get all the questions right, so you don't learn anythi ng about the distribution within those bits of the tail.

Testing millions of people with one or more average-centred tests doesn't t ell you any more about the contents of the tails that the tests don't sprea d out.

One always expects a Gaussian distribution. You've got to do measurements t o confirm that expectation.

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Flynn effect. IQ test results are rising. Flynn thinks that a lot of it is our persuading the entire population to think in ways that suit IQ tests, b ut scoring high on an IQ test improves your chances of getting a good educa tion, a good job and a good income, and thus your chance of reproducing, so there may be an element of evolution in action too.

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It's not the "single most useful measure". Secondary school students and un dergraduates are are tested with written examinations, which take longer to do and are more expensive to mark. IQ tests haven't replaced them, and nev er will.

Graduate students have to write a thesis which is then examined by a bunch of experts, which is an even more tedious and expensive process.

The more interesting question about IQ tests - granting the enthusiasm of t heir proponents - is why they haven't replaced the more tedious and expensi ve traditional measures.

Academics who specialise in IQ testing do seem to think that it's useful, b ut we've also got osteopaths, homeopaths and theology faculties.

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Bill Sloman, Sydney

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There are? That's news. The last stat I read about gender and math ability was something from the MAA where it was found females on average are better than males at math, but the spread on male performance was much greater, a nd in particular, the exceptionally good males were way beyond anything see n in the female population.

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emember that while Blacks may have a lower mean IQ , there are a lot of ver y high IQ blacks. It other words you have to judge individuals.

y was something from the MAA where it was found females on average are bett er than males at math, but the spread on male performance was much greater, and in particular, the exceptionally good males were way beyond anything s een in the female population.

OK sorry, my mistake. I have no idea who is better at math.

George h.