The importance of remembering that scientists are not mathematicians

I’ve been reading Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem, by Simon Singh.  Normally, I’d shy away from a book like this — after all, it’s about math! — but it was required reading for my book club, and it’s proven to be delightful.  To the extent there is math in it, Singh masterfully simplifies complex ideas so that even math illiterates like myself can understand them.  Indeed, I suspect that, if I’d had a teach like Singh when I was in school, one who teaches why something matters, or how it came to be, rather than just demanding that one memorize meaningless formulas, I might not be the math illiterate (and math phobe) that I am today.

But my ruminations about books and math aren’t actually why I’m writing right now.  Instead, I wanted to comment on the different types of thinking in the sciences.  I’m ashamed to admit that I never really sat down and analyzed the different intellectual approaches people on the “science side” use.  To me, the world was binary:  science mind (including math) and not science mind (including me).  Sure I knew that engineers could be a bit obsessive compulsive, but it was a trait I admired, so I never thought more about it.

What never occurred to me, however, is that specific branches of science demand different approaches to finality — or, as it’s called in math, “absolute proof.”  Let me have Singh describe this concept.  I’ll quote at some length from his text at pages 20-22 (in the hard copy version 0f his book):

The story of Fermat’s Last Theorem revolves around the search for a missing proof. Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as understood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras. The idea of a classic mathematical proof is to begin with a series of axioms, statements that can be assumed to be true or that are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.

Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value of such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favor of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.

However, the scientific theory can never be proved to the same absolute level of a mathematical theorem: It is merely considered highly likely based on the evidence available. So-called scientific proof relies on observation and perception, both of which are fallible and provide only approximations to the truth. As Bertrand Russell pointed out: “Although this may seem a paradox, all exact science is dominated by the idea of approximation.” Even the most widely accepted scientific “proofs” always have a small element of doubt in them. Sometimes this doubt diminishes, although it never disappears completely, while on other occasions the proof is ultimately shown to be wrong. This weakness in scientific proof leads to scientific revolutions in which one theory that was assumed to be correct is replaced with another theory, which may be merely a refinement of the original theory, or which may be a complete contradiction.

I know that, having read that, you’re thinking exactly what I’m thinking:  Global Warming.  You’re thinking of falsified data, of non-vanishing glaciers, of robust polar bear populations, and of the other cascade of data showing wrong-headed theories supported by bad, careless, or out-and-out fraudulent “science.”  Credulous people, ideologically driven people, and people who confuse scientific theory with the absolute proof of a mathematical theorem were willing to accept that “the science is settled.”  But unlike math, which can see a theorem being finally and definitively proved, real science is never settled, and anyone who claims that must be a liar.

Certainly, we know that some scientific theories are more stable than others, and we’ve built large parts of our world on that.  But when people purport to take the dynamics of the sun, the moon, the earth and predict the climate outcome years or even decades in advance, and then it turns out that they’ve done so entirely without regard to the sun, the moon, and the earth, you know you’ve got mysticism and faith, and nothing remotely approaching science, let alone the sureties of math.

I’ll leave you with a joke, also from Singh’s book, although it originally comes from Ian Stewart, in his book Concepts of Modern Mathematics:

An astronomer, a physicist, and a mathematician (it is said) were holidaying in Scotland.  Glancing from a train window, they observed a black sheep in the middle of a field.  “How interesting,” observed the astronomer, “all Scottish sheep are black!”  To which the physicist responded, “No, no!  Some Scottish sheep are black!”  The mathematician gazed heavenward in supplication, and then intoned, “In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.”

Since you’re all much cleverer than I at jokes and bon mots, I’ll leave you to imagine what the AGW “scientist” would have said upon seeing that sheep in that field.

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  • BobK

    The AGW ‘climatologist’:
    “The science is now settled.  Carbon is black, the sheep are black.  These FACTS are clearly indicative of an alarming trend among domesticated oppressed fellow mammals.  Unless we take drastic remedial action NOW, within 50 years ALL sheep worldwide will be both black and Scottish!”

  • gpc31

    Great joke at the end!

    Here is another difference between pure math and applied science:  Much of natural science is pragmatic, going back to its Baconian origins (knowledge is power, the conquest of nature and the relief of man’s estate, etc.).  Therefore, the idea of what counts as a significant fact in the first place depends on what you are trying to do.  There is a great deal of social direction in science (certainly in the funding thereof).

    Scientists try to game the system in terms of funding.  The best example I heard of was on the belmont club site.  One scientist remarked that while you couldn’t get funding to research the feeding habits of squirrels, getting money to study the effects of global warming on the feeding habits of squirrels was a sure thing.  Thus a phony consensus is born.  Not nefarious, but not innocent, and definitely insidious.

    Max Planck, a great physicist, once said that “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”  I am not sure that I entirely buy that interpretation, but I do believe that the left tried to speed up the process by screaming “the science is settled” and by indoctrinating the young about AGW, and that I bitterly resent.

  • JKB

    The climatologist pulled out his laptop and started writing a grant proposal to study “The Heat Carbonization of in-situ Wool due to Man-caused CO2 Build-up over Scottish Grazing Sectors.”
    Interesting observation that explains a lot about the adamant belief in “experts” by non-scientists.  Those trained in science know nothing is ever “settled” even those concepts considered fundamental.  Much less, data analysis that hasn’t been subjected to open investigation and predictions that have not yet come to pass.
    But then desires for funding and uncurious minds can cause even science-trained individuals to deny this fact.  I once had a M.S. in Oceanography adamantly refused to believe that the Equatorial Current had diurnal components even in the face of readouts from the Acoustic Doppler Current Profiler and the fact the ship was tracking very well on the drift plan I developed using the set and drift measured.  Apparently, he couldn’t get past what he’d learned in a textbook and face what was in front of his lying eyes.

  • roylofquist

    There are three kinds of mathematicians. Those who can add and those who can’t.

  • Oldflyer

    Al Gore would observe that anyone could see that the sheep’s wool was shorter than the wool of sheep 2,000 years ago, as observed by studying the preserved dung found in “selected” parts of the world, therefore proving that global warming not only exists, but the entire sheep population will be drowned when the seas rise.

    Would it be fair to say that Climatologist (trained and untrained) have used data of questionable certainty, to develop a hypothesis of dubious validity; then they developed computer models of  unproven ability to make predictions with absolute certainty?

    I have a minimal  knowledge of computer modeling, through studies leading to my MS  in the uncertain field titled “Computer Systems Managment”.   I know that modeling is a useful tool, but has some pretty serious limitation; especially if the input data is uncertain, and the intial assumptions are not completely valid.  After some recent reading about AGW, I challenged a friend to project on a scale of 1 to 10 the liklihood of developing a model for a system as complex as the climate, and using to predict future trends, and consequences of the trends. 

    Now this fellow is one of the smartest,  most cerebral people I know.  He would be called Dr, if he hadn’t wasted so much of his youth flying airplanes.   We were squadron mates and roomates aboard ship; he holds an MS in Operations Research; was a Navy test pilot; and post-USN worked for a think tank  performing analysis of complex problems.  He,of course, used modeling as a tool.  In response to me, he assigned a number of 1 to 2 to a model that replicated the climate.  He did say that he would assign a 3 or  4  to a model that predicted  trends over the next 50 or 100 years because he feels there is enough data.   My friend mentioned that his son, whom he considers to  much smarter than he, assigned a top score of 1 to any aspect of climate modeling.

    He went on to relate an  interesting  personal antecdote  to illustrate the the problem of modeling complex problems (not nearly on the scale of the climate ,of course, in which you know there are a large number of unknowns (not to mention Rumsfeld’s dreaded unknown–unknowns)).

    My friend was the lead on an analysis to predict the number of aircraft losses to be expected in a war with Iraq (the first Gulf War).  His staff included PHDs from MIT, CALTECH and Stanford; so there was no shortage of brain power.  In addition to his own intellectual capabilities he had the practical experienc of  actually going  to war over North VietNam.  They predicted fairly horrendous losses, and he dutifully briefed the JCS and Congressional committees accordingly.   Fortunately, when events played out they had overestimated the losses by a factor of 20x or more.  One unforseen factor that played into the error was that the Iraqi air defense system shut down their radars when they felt threatened; and they were always threatened.  The U.S. anti-radiation missile attack aircraft used the names of beers as their radio call signs; a fact the Iraqis picked up.  The Americans quickly figured this out and thereafter all strike aircraft used a “beer” call sign and successfully shutdown the air defense radars when they entered the area simply by talking on the radio.

    They  refined models to the point that they were pretty  accurate in the 2nd gulf war.  Still, this problem, though many orders of magnitude less complex than the earth’s climate, illustrates some of the pitfalls in predictive modeling.

  • March Hare

    My favorite mathematician/engineer joke (I’ve know a few of both–and, yes, this joke dates from when most mathematicians and most engineers were male):
    Put a beautiful woman at the end of a hall with a mathematician and an engineer at the other end.  Tell them they can only move half the remaining distance to get to the woman.
    The mathematician states that he will never reach the woman.
    The engineer states that he will get close enough for all practical purposes.

  • David Foster

    JKB says: “Apparently, he couldn’t get past what he’d learned in a textbook and face what was in front of his lying eyes.” In his novel “That Hideous Strength,” C S Lewis describes his protagonist, a sociologist, thusly: “..his education had had the curious effect of making things that he read and wrote more real to him than the things he saw. Statistics about agricultural labourers were the substance: any real ditcher, ploughman, or farmer’s boy, was the shadow…he had a great reluctance, in his work, to ever use such words as “man” or “woman.” He preferred to write about “vocational groups,” “elements,” “classes,” and “populations”: for, in his own way, he believed as firmly as any mystic in the superior reality of the things that are not seen.”

  • Gringo

    From a family friend, from decades ago. Various professionals were arguing about whether 6 was a prime number or not. The philosopher said, “It is up to 6 to decide for itself whether it was a prime number or not: it has free choice.” The journalist said, “It depends on whether 6 is in the prime of life or not.” The statistician said, “1 is not a prime , 2 is a prime , 3 is a prime n, 4 is not a prime , 5 is a prime, so the odds are that 6 is a prime number.” The engineer said, “ 1’s a prime, 2’s a prime, 3’s a prime, 4’s a prime, 5’s a prime..”
    Q: How does a mathematician induce good behavior in her children? A: `If I’ve told you n times, I’ve told you n+1 times…’
    I was indifferent to math until high school. I had a very poor  math teacher in 9th grade- a family friend coincidentally. It made no difference. The math program was so good- a New Math (UICSM) program that emphasized proofs- that I  taught myself math from the book. Math became my favorite subject.


    The AGW scientist would take five years of units of sheeps’ wool harvested per hundred acres, successively subtract 5% from each year’s yield. Assume the same number of sheep per acre.
    500 Y ear 1
    525 Year 2
    470 Year 3
    510 Year 4
    505 Year 5
    The data would then be adjusted thusly
    500 Year 1
    525 Year 2 less 5%
    470 Year 3 less 10 %
    510 Year 4 ess 15%
    505 Year 5 less 20% to come up with the following data
    500 Year 1
    499 Year 2
    425 Year 3
    434 Year 4
    404 Year 5
    The reduced yield in pounds of wool per acre is an indication of global warming, as the sheeep do not need as much wool to keep warm.

  • Bookworm

    I knew I could trust all of you to come up with good AGW responses to that black sheep.

    As for me, I was plagued by truly vile math teachers throughout my years at public school.  This isn’t just me spewing sour grapes.  Each of my teachers, without exception had a horrible reputation.  Add to this the fact that I had no natural aptitude for math, and you can imagine the scale of my academic failings in that subject.  Ironically, I’ve discovered as an adult that, if someone explains principles clearly, I can grasp them and even apply them.  A good teacher would have made a world of difference to me.  But that’s water under the bridge, and I excel at other things.  All is good.

  • riw777

    It’s essentially the difference between deductive and inductive reasoning; because of formal logical proofs, math, which is based on inductive reasoning, is always more provable than science, which is based on deductive reasoning. In the real world, we tend to trust deductive rather than inductive reasoning–which is why, I think, people have such a hard time with math in general (including me).
    In reality, I think both are needed to describe the real world; although they battle constantly, neither side wins the war. I also tend to think good engineers “know,” through developed practice, which to use where to arrive at the “right” answer. That’s what makes them different from scientist and mathematicians, in the end. There’s an “anchor to reality” that keeps engineers honest–if I design a network that kills people, I would lose my status as an engineer.