Essay by Charlie Martin
We know, with great certainty, that the overall average temperature of the Earth has warmed by several degreees in the last 400 years, since the end of the Little Ice Age. Before that was a period called the Medieval Warm Period; before that was another cold period, and back at the time of the Romans there was a long period that was significantly warmer — Southern Britain was a wine-growing region. What we’re a lot less certain about is why?
Of course, the “why?” here has been, shall we say, pretty controversial. It’s worth wondering about the controversy and about the social mechanisms through which science is done — I wrote about them during the Climategate controversy as the “social contract of science” — but that’s not what I want to talk about today. Instead, let’s talk about how a scientist thinks about these sorts of questions and arrives at new answers. Back in grad school we called that “doing science”, and it was something everyone liked doing and wished they could be doing instead of whatever they actually were doing, like faculty meetings and refereeing papers.
The process of “doing science” is something you usually learn more or less by osmosis, but there are some good hints around. One of the best is a paper from the 16 October 1964 issue of Science, “Strong Inference” by John R Platt. Let’s say we have some phenomenon of interest, like global warming, or high blood sugar, or that damned yellow patch in my lawn. We want to know why it happens. Platt’s strong inference describes the process we should use when “doing science” as:
- We generate a number of alternate explanations, hypotheses, that might explain the phenomenon.
- For each hypothesis, we come up with an experiment which will prove the hypothesis wrong. That is, not one that “proves the hypothesis”, but one which, if successful, would disprove or falsify the hypothesis. (Sir Karl Popper argued in his book The Logic of Scientific Discovery that this falsification was the core of scientific knowledge.)
- We do the experiments. If an experiment falsifies a hypothesis, we discard it ruthlessly. Then we go back to (1) and try again.
A lot of times, the rub — and the really creative thinking — comes in from finding the right experiment. Richard Feynmann was known for an ability to see right through a problem to a simple and elegant experiment that would disprove a hypothesis. He demonstrated this during the review following the Challenger disaster. You may remember that the launch happened on a very cold morning in January; less than two minutes after launch the Space Shuttle Challenger blew up, killing all seven astronauts.
The question, as always, was “why?”
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“””””…..Rabe says:
June 26, 2013 at 12:21 pm
Mr. Garcia, please… you didn’t read what Fermat wrote. Do it, read it aloud. Then look at the amount of paper Mr. Wiles’ proof would have needed in Fermat’s time, if the mathematical proofs and methods used by Wiles would have been known back then. Do you find Fermat’s description to be appropriate? So I think what Mr. Smith wrote is correct……”””””
It’s always nice to know that some people actually read what I write; instead of what they think I meant. Unless I make a typo, I always mean exactly what I write.
But Rabe, have you ever considered another intriguing possibility, about the brief note Fermat wrote in his margin. On my first reading of that “tweet”, the message I got, was much more dramatic, than the fact his proof was less that 250 close spaced typed pages.
I believe Fermat’s proof was so simple, that he simply didn’t bother to write it down, thinking any 4-H clubber would figure it out for himself, after simply being made aware of the theorem.
That’s what rattles my cage thinking of Fermat’s theorem. He did say he had a remarkable / wonderful / whatever proof of the conjecture. He was not known for telling falsehoods.
The thought that nobody has ever stumbled over whatever proof he had, and then discovered a flaw in it.
I’m not aware of ANY history of simple “proofs”, subsequently dismissed. How many known “proofs” later discredited, exist in the math literature, before Wiles gargantuan tome.
I once developed my own version of Cardan’s solution for the roots of a cubic polynomial, which is purely algebraic, as distinct from the trigonometrical solution. If I recall how I did it, Cardan’s solution only gives the roots in terms of complex numbers, so you can’t get the numerical value of them. The trig solution yields at least the one real root, which, you can divide out, and then solve the quadratic, for the real or complex pair.
As for quardics, I studied Ferrari’s solution till all my hair fell out, and never did get the gist of it.
george .e. smith says:
June 27, 2013 at 12:40 am
“””””…..Rabe says:
June 26, 2013 at 12:21 pm
Mr. Garcia, please… you didn’t read what Fermat wrote. Do it, read it aloud. Then look at the amount of paper Mr. Wiles’ proof would have needed in Fermat’s time, if the mathematical proofs and methods used by Wiles would have been known back then. Do you find Fermat’s description to be appropriate? So I think what Mr. Smith wrote is correct……”””””
It’s always nice to know that some people actually read what I write; instead of what they think I meant. Unless I make a typo, I always mean exactly what I write.
But Rabe, have you ever considered another intriguing possibility, about the brief note Fermat wrote in his margin. On my first reading of that “tweet”, the message I got, was much more dramatic, than the fact his proof was less that 250 close spaced typed pages.
I believe Fermat’s proof was so simple, that he simply didn’t bother to write it down, thinking any 4-H clubber would figure it out for himself, after simply being made aware of the theorem.
That’s what rattles my cage thinking of Fermat’s theorem. He did say he had a remarkable / wonderful / whatever proof of the conjecture. He was not known for telling falsehoods.
Andrew believes that Fermat had fooled himself into thinking that he had a proof. He says that the proof is much too long (150 pages) and uses many modern mathematical techniques which weren’t known in Fermat’s time. It took Andrew about 7 years to develop the proof!
“””””…..Phil. says:
June 27, 2013 at 1:20 pm
george .e. smith says:
June 27, 2013 at 12:40 am
“””””…..Rabe says:
June 26, 2013 at 12:21 pm
Mr. Garcia, please… you didn’t read what Fermat wrote.
……………………..
……………………..
…..”””””
Well Phil, I take nothing at all away from Andrew Wiles, and his proof.
Actually, if my memory serves me, Andrew’s paper does not directly prove Fermat’s theorem. Instead it proves some other exotic mathematical theorem, which if true, then leads to Fermat’s theorem.
And I’m just taking a WAG, but I suspect, that, in Fermat’s day, that other esoteric theorem, was entirely unknown to mathematics..
So if that conjecture, whose name escapes me was known in Fermat’s day to be (if true), a direct path to the solution of Fermat’s theorem; then presumably, Fermat can only be faulted, for failing to find Mr Wile’s 150 page proof of that other conjecture.
But I think it is presumptuous to simply dismiss Fermat, as having “fooled himself”.
So I would repeat my question. How many “Proofs” of Fermat’s last theorem, are known, that were later discarded, as having “simply fooled” their authors; prior to Andrew Wile’s proof of the associated conjecture. (about which, I plead total ignorance; well beyond my pay grade.).
I have a similar queezy feeling about an entirely unrelated historical matter.
When some years ago, the well preserved body of George Lee Mallory, was discovered, where he fell, (to), on Mt Everest, it started a whole new detective story, about whether, Mallory and Irvine made it to the top, in 1924, or whether they fell, before reaching the top, or even, were they going up or down when they fell.
So eventually Sir Edmund Hilary, was asked what he thought about the question. He is reported to have said, it doesn’t matter whether they made it or not. The aim was to make it back alive, as well as make it to the top. True enough, RF Scott, is often criticized for his fateful, and fatal, and poorly planned South Pole failure, while deservedly Amundsen is lauded for his success.
So I was somewhat disappointed in Hillary’s answer. He could have said: ‘Well a lot of romantics secretly hoped that they made it.’ which in no way would have diminished his, and Tenzing’s success.
So all praise for Andrew Wile for his solution that simply was not in any way available in Fermat’s day. But what if someone else, stumbles over Fermat’s proof that was too simple to bother writing down.
The Roman Warm Period was not the most recent time when England grew grapes and made wine. That was done in some fairly recent years.
Also, the linked article has a graph that does not support its claim of lack of warming for 17 years. It merely shows warming largely stagnant for the past ~11-12 years, and pushing the bottom edge of the 95% confidence range of the model consensus.
Also, that article claims that temperatures during the Little Ice Age were several degrees cooler than now. The variation appears to me about 1.8 degrees C according to a Loehle study published in Energy and Environment. That study is favored by Dr. Roy Spencer, who is somewhat on the skeptical side, and a climate scientist.
Of course we are affecting. If we are continuing urbanization, deforestations and desert expanding global warming / climate change will be more and more eminent. Moisture contents on the surface of the earth controls our climate. We have changed the soil surface into concrete surface thus reduced the evaporation and water absorption system of nature. For details please click on my name.
Of course we are affecting. If we are continuing urbanization, deforestations and desert expanding global warming / climate change will be more and more eminent. Moisture contents on the surface of the earth control our climate. We have changed the soil surface into concrete surface thus reduced the evaporation and water absorption system of nature. For details please click on my name.
________________________________
george e. smith says:
June 27, 2013 at 3:30 pm
“””””…..Phil. says:
June 27, 2013 at 1:20 pm
george .e. smith says:
June 27, 2013 at 12:40 am
“””””…..Rabe says:
June 26, 2013 at 12:21 pm
Mr. Garcia, please… you didn’t read what Fermat wrote.
……………………..
……………………..
…..”””””
Well Phil, I take nothing at all away from Andrew Wiles, and his proof.
Actually, if my memory serves me, Andrew’s paper does not directly prove Fermat’s theorem. Instead it proves some other exotic mathematical theorem, which if true, then leads to Fermat’s theorem.
Yes George, the Taniyama-Shimura conjecture (theory). It was the link between this and Fermat that inspired Andrew to take up the proof again, he realized that if he proved T-S he also proved Fermat.
And I’m just taking a WAG, but I suspect, that, in Fermat’s day, that other esoteric theorem, was entirely unknown to mathematics..
Yes T-S originated in the 50’s I think.
So if that conjecture, whose name escapes me was known in Fermat’s day to be (if true), a direct path to the solution of Fermat’s theorem; then presumably, Fermat can only be faulted, for failing to find Mr Wile’s 150 page proof of that other conjecture.
But I think it is presumptuous to simply dismiss Fermat, as having “fooled himself”.
I think given the work that Andrew did on Fermat (not just on the T-S approach) he’s probably the best qualified to make that judgement, and that’s certainly what he said.
So I would repeat my question. How many “Proofs” of Fermat’s last theorem, are known, that were later discarded, as having “simply fooled” their authors; prior to Andrew Wile’s proof of the associated conjecture. (about which, I plead total ignorance; well beyond my pay grade.).
At least one, Andrew’s first proof turned out the have a flaw which took another year to fix, so he knows about being fooled!
@ur momisugly Phil”
“”””…..I think given the work that Andrew did on Fermat (not just on the T-S approach) he’s probably the best qualified to make that judgement, and that’s certainly what he said……”””””
Well Phil, I’m presuming that you know the chap, so that you have some idea of the extent to which he studied Fermat; pre and post T-S.
So that leads me to make the following conjecture; aka WAG.
Given that Fermat knew nothing of the (non) existence of the T-S conjecture, and how it would come to relate to his last theorem. And given that Fermat BELIEVED that he had a proof of his theorem (that was not a 150 page proof of T-S). And given that Andrew Wile extensively studied Fermat’s theorem, prior to T-S, so can be regarded as an (if not the) expert.
Then it is reasonable to assume that in his studies, he must have also discovered Fermat’s simple but fallacious proof..
So for me; while congratulating T-S and Andrew Wile, on their combined success; the big outstanding problem is :
NOBODY HAS DISCOVERED FERMAT’S “PROOF” OF FERMAT’S LAST THEOREM.
I think that is every bit as important as proving the theorem, and would surely have been found by anyone, knowledgeable enough to believe Fermat fooled himself.
PS My formal mathematics studies, only went to MSc (in Physics & Maths) level, and I did not take the finals.
So T-S is quite beyond my horizon of understanding. So I am quite certain, that I could read Andrew Wiles 150 page solution, along with T-S; and have not the foggiest idea what they were talking about.
Good question on falsification Charile!
One way of testing the greenhouse gas hypothesis and the size of the effect is the following:
Since the 1970s, satellites have been measuring the heat emerging from our atmosphere into space. If no reduction is observed in heat emitted at wavelengths associated with absorption by GHGs , the theory would be nicely falsified. In fact, a pronounced dip in energy at just those wavelengths is clearly observable, a dip that has been growing steadily in tandem with the rise in atmospheric GHG concentration…I don’t know why we don’t hear more about these experiments, there is a good overview at
http://www.skepticalscience.com/empirical-evidence-for-co2-enhanced-greenhouse-effect.htm
“””””…..cormac says:
June 28, 2013 at 1:06 pm
Good question on falsification Charile!
One way of testing the greenhouse gas hypothesis and the size of the effect is the following:
Since the 1970s, satellites have been measuring the heat emerging from our atmosphere into space. If no reduction is observed in heat emitted at wavelengths associated with absorption by GHGs , the theory would be nicely falsified. In fact, a pronounced dip in energy at just those wavelengths is clearly observable, a dip that has been growing steadily in tandem with the rise in atmospheric GHG concentration……..”””””
Did you read, what you just wrote Cormac ?
First off, from a ‘satellite in outer space’, it is not possible to observe ANY HEAT emerging from earth for essentially none is.
What is being emitted, is LWIR EM radiation energy, which is NOT “heat”.
According to Kevin Trenberth’s cartoon budget, ONLY 40 W/m^2 out of the 390 W/m^2 emitted from earth surface, escapes to outer space. The rest is captured by the atmosphere, including GHGs.
Also according to the overwhelming consensus (maybe 97%) of all scientists believe that ordinary atmospheric gases (N2, O2, Ar) do not emit LWIR thermal EM radiation; only GHGs can do that.
So the earth spectrum from outer space is supposed to consist of a weak (40 W/m^2) surface emitted thermal spectrum, plus a whopping great GHG band spectrum, because the atmosphere can only cool by GHG resonance emissions.
So when you talk of “a pronounced dip in energy at just those wavelengths”, that is an observation (presumably), that flies in the face of assertions that only GHG spectra are emitted by the atmosphere. An increase in GHGs, would necessarily result in an INCREASE in the GHG bands peaks, NOT a decrease.
So there must be something wrong with the assertion that the atmosphere can ONLY emit GHG band spectra emissions, and NOT broad thermal radiation spectra. Apparently the overwhelming consensus of 97% of all scientists is in error, ant the atmospheric gases, must be able to radiate broad thermal spectra..
How can you tell it is the gas causing that effect. Can you make GH out of gases? If yes show me how? Gases are freely moving molecules; they are helping the earth to cool down by convection method of heat transmission. Heat is absorbed by the concrete jungles we are creating to cover the soil surface of the earth. The soil surface holds water, allows water to recharge ground water (thus reducing draining water to the sea), evaporation of water is most effective cooling system in Nature (concrete is dry so this effect is reducing thereby increasing temperature of the Earth, the GW); convection method of heat transmission is not enough to cool the earth as what we think of – so transmission of heat to the space is less than we expect, since heat is held by the concrete jungles. Thus, of course, human is affecting but not due to gases but by urbanization – homes, roads, pavements, deforestations, deserts expansion etc. I suppose you know what is a Green House? No gases can be ‘green house gases’ – they are SOLID AND TRANSPARENT LIKE PLASTICS AND Glasses. For details on CC and GW please click on my name.