Well, the quote could be interpreted as “Any scientific theory must ultimately produce some numbers, so that reality can be measured and we can see whether the numbers match.”
Another interpretation is “A scientific theory ultimately isn’t a scientific theory at all unless it’s essentially a set of equations.”
Counterfactual: the theory of evolution is one of the most successful scientific theories, yet it contains no equations; nor numbers. It is rather a framework of ideas in which observations can be made sense of.
I tend to agree with you that numbers are inessential in a scientific theory, and that Darwin’s theory is a good example of this. But your critics also have a point that some nice math has been added to the theory since Darwin’s time. (Not enough of a point to justify downvoting you, though, IMHO).
As a smaller scale example of a non-numerical scientific theory, consider the theory that the historical branching order of the Great Ape family tree is “First orangutan, then gorilla, then man, leaving the two species of chimp.” That is a meaningful and testable scientific theory as it stands, even though there are no numbers involved. But what spoils my example a little is the observation that this theory is improved by adding numbers. “Orangutan branched ~12M years ago, gorilla 6M, man 5M, bonobo 0.5M.”
This does highlight a problem in the insistence on numbers, though. What’s required is not numbers but mathematics, something we can formalize. Classical theories dealt largely in real numbers and functions of real numbers but there’s nothing wrong with a theory we get trees out of instead of numbers. (Of course, we can then use numbers in describing those—numbers are enormously useful—but they don’t need to be the direct result.)
the historical branching order of the Great Ape family tree is “First orangutan, then gorilla, then man, leaving the two species of chimp.” That is a meaningful and testable scientific theory as it stands, even though there are no numbers involved.
Well, the numbers 1,2,3 do show up implicitly here, in the ordering.
The theory of evolution was discredited around, IIRC, 1900, because the math didn’t work out, because people didn’t know genes were discrete, and thought they were analog. It was resurrected after people learned genes were discrete, and found the math worked.
(I haven’t looked at this math myself, so I could be wrong.)
There is some minor confusion here that it may be worth clearing up. The mathematical ‘disproof of Darwin’ you seem to be thinking of was the work of Fleeming Jenkin who wrote to Darwin with his objections around 1870.
Jenkin’s argument was based on the reasonable-at-the-time assumption of ‘blending inheritance’, the idea that the features of an organism (height, say) should simply be the average of the features of its parents, plus or minus a random perturbation. Jenkin showed that if this were how heredity worked, then natural selection would be almost completely ineffective.
Darwin was troubled by Jenkin’s argument, in part because he did not understand the math. One of his correspondents commiserated:
The mathematicians must be a singularly happy race, seeing that they alone of men are competent to think about the facts of the cosmos. … Mathematics are … the sciences of number and measurement, and as such, one is at a loss to perceive why they should be so essentially necessary to enable a man to think fairly and well upon other subjects. But it is, as you once said, that when a man is to be killed by the sword mathematical, he must not have the satisfaction of even knowing how he is killed.
Jenkin’s objections were never all that influential, because no one else understood the math either. But the rediscovery of Mendel around 1900 provided the needed correction to the ‘blending inheritance’ assumption. Fisher ‘did the math’ refuting Jenkin around 1920, and republished his argument as the first chapter of his book in 1930. Available online and definitely worth a read.
Unfortunately, the wikipedia article on Jenkin confuses his ‘refutation’ of Darwin with that of William Thompson (Lord Kelvin) who wrote in 1897 that the earth was “was more than 20 and less than 40 million year old, and probably much nearer 20 than 40”. His argument was based on how long it would take for the core of the earth to cool down to its current (roughly calculable) level. Lord Kelvin’s math was right, but he failed to take into account the heating effects of radioactive decay. Radioactivity was first discovered in 1895 and was still not well understood in Kelvin’s time. Rutherford finally disposed of this ‘mathematical refutation of Darwin’ around 1910.
Unfortunately, the wikipedia article on Jenkin confuses his ‘refutation’ of Darwin with that of William Thompson (Lord Kelvin) who wrote in 1897 that the earth was “was more than 20 and less than 40 million year old, and probably much nearer 20 than 40”.
The theory of evolution was discredited around, IIRC, 1900
The only source I found semi-supporting that comes from a document which reads strangely, as if it is teaching the “controversy”. Look at the wording:
By 1900, natural selection had been so discredited that few scientists accepted it as the mechanism of evolution. By all accounts, however, they all accepted the so-called “fact” that species evolve.
Richard Bellman, “Eye of the Hurricane”
Sounds like a traditional-rationality precursor to “hypotheses are expectation-constrainers”.
It doesn’t sound like that to me. Can you elaborate?
Well, the quote could be interpreted as “Any scientific theory must ultimately produce some numbers, so that reality can be measured and we can see whether the numbers match.”
Another interpretation is “A scientific theory ultimately isn’t a scientific theory at all unless it’s essentially a set of equations.”
I agree, although I think the second sentence (“Theories stand or fall, ultimately, upon numbers”) is sufficient to justify the former interpretation.
Counterfactual: the theory of evolution is one of the most successful scientific theories, yet it contains no equations; nor numbers. It is rather a framework of ideas in which observations can be made sense of.
Price’s Equation? Fisher’s fundamental theorem? Hardy-Weinberg law?
I tend to agree with you that numbers are inessential in a scientific theory, and that Darwin’s theory is a good example of this. But your critics also have a point that some nice math has been added to the theory since Darwin’s time. (Not enough of a point to justify downvoting you, though, IMHO).
As a smaller scale example of a non-numerical scientific theory, consider the theory that the historical branching order of the Great Ape family tree is “First orangutan, then gorilla, then man, leaving the two species of chimp.” That is a meaningful and testable scientific theory as it stands, even though there are no numbers involved. But what spoils my example a little is the observation that this theory is improved by adding numbers. “Orangutan branched ~12M years ago, gorilla 6M, man 5M, bonobo 0.5M.”
This does highlight a problem in the insistence on numbers, though. What’s required is not numbers but mathematics, something we can formalize. Classical theories dealt largely in real numbers and functions of real numbers but there’s nothing wrong with a theory we get trees out of instead of numbers. (Of course, we can then use numbers in describing those—numbers are enormously useful—but they don’t need to be the direct result.)
Well, the numbers 1,2,3 do show up implicitly here, in the ordering.
The theory of evolution was discredited around, IIRC, 1900, because the math didn’t work out, because people didn’t know genes were discrete, and thought they were analog. It was resurrected after people learned genes were discrete, and found the math worked.
(I haven’t looked at this math myself, so I could be wrong.)
There is some minor confusion here that it may be worth clearing up. The mathematical ‘disproof of Darwin’ you seem to be thinking of was the work of Fleeming Jenkin who wrote to Darwin with his objections around 1870.
Jenkin’s argument was based on the reasonable-at-the-time assumption of ‘blending inheritance’, the idea that the features of an organism (height, say) should simply be the average of the features of its parents, plus or minus a random perturbation. Jenkin showed that if this were how heredity worked, then natural selection would be almost completely ineffective.
Darwin was troubled by Jenkin’s argument, in part because he did not understand the math. One of his correspondents commiserated:
Jenkin’s objections were never all that influential, because no one else understood the math either. But the rediscovery of Mendel around 1900 provided the needed correction to the ‘blending inheritance’ assumption. Fisher ‘did the math’ refuting Jenkin around 1920, and republished his argument as the first chapter of his book in 1930. Available online and definitely worth a read.
Unfortunately, the wikipedia article on Jenkin confuses his ‘refutation’ of Darwin with that of William Thompson (Lord Kelvin) who wrote in 1897 that the earth was “was more than 20 and less than 40 million year old, and probably much nearer 20 than 40”. His argument was based on how long it would take for the core of the earth to cool down to its current (roughly calculable) level. Lord Kelvin’s math was right, but he failed to take into account the heating effects of radioactive decay. Radioactivity was first discovered in 1895 and was still not well understood in Kelvin’s time. Rutherford finally disposed of this ‘mathematical refutation of Darwin’ around 1910.
OMG! I checked. It is true! What a mess!
The only source I found semi-supporting that comes from a document which reads strangely, as if it is teaching the “controversy”. Look at the wording:
The so-called “fact”!