** “In school our children are being taught two conflicting systems of mathematics. In primary school they learn to calculate in an empirical way: one apple and one apple makes two apples. But later on they they are told that is wrong, and learn some formal mathematics where you postulate some axioms and use the deductive method to arrive at conclusions from the axioms. This makes matters very complicated: Whitehead and Bertrand Russell took 368 pages to deductively prove 1+1=2 in their book. Decolonised mathematics eliminates this needless complexity and accepts the natural and empirical way; it is simple and easy.”**
**
Indeed, that is how I learned mathematics using deductive proofs. The famous example is: “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” So what is wrong with that method of deduction? Why is this Eurocentric, apart from the fact that a male is used in this example?
Raju: “First of all, I don’t use the term ‘Eurocentric’ because it wrongly suggests that a massive piece of deliberate mischief was an innocent mistake. Second there is nothing wrong with the method of deduction as such, which was used also, for example, in India. Of course, attributing this syllogism to Aristotle is the usual false Western history: there is nil evidence to link the syllogism to Aristotle. What is uniquely Western and wrong are the claims that (a) deduction is infallible, (b) that it is universal (c) that deductive proof is superior to empirical proof, and that (d) it is possible to arrive at valid knowledge without any empirical inputs, as in formal mathematics. All these wrong claims lead to the wrong belief that Western (formal) mathematics is superior and the only right way to do mathematics.”**
“Empirical proof is rejected by Western mathematics on the grounds that empirical proof is fallible. Our senses might mislead us. To use a classical example from Indian philosophy: I might mistake a rope for a snake or a snake for a rope. But deductive proof too is fallible: one may easily mistake an invalid deductive proof for a valid one. For example, the very first proposition of “Euclid’s” Elements has an invalid deductive proof. But for 8 centuries that book was mistakenly regarded by all the foremost minds in the West as the model of deductive proof, when, in fact, there isn’t a single valid deductive proof in it, as Bertrand Russell too emphasized. How do you know that his own 368 page proof of 1+1=2 is valid? You just blindly trust authority, and such blind trust can be very fallible. Empirical proofs are never so fallible: one might mistake a rope for a snake, but the Western error about “Euclid’s” Elements, is like mistaking a rope for an elephant.”
**
“No, on the contrary they are inferior. Divorced from the empirical, even a valid deductive proof does not lead to valid knowledge or even to approximately valid knowledge,” states Raju. “Using the deductive method any silly proposition whatsoever can be proved as a mathematical theorem from some postulates.”**
to summarize, he is arguing that Western Mathematics reliability on the Deductive method is inferior to the empirical method
You can’t do “empirical” math. Go ahead, observe the square root of two “in nature.” Ain’t gonna happen.
Now, I do know some people who insist on “constructive math.” They refuse to accept any number that doesn’t represent an actual, physically real number of things. Billions? Sure, there are billions of people. Trillions? Sure, atoms and molecules and things. But, say, 10^800? “That is not a number!”
That was straight word salad to me. Is the type of mathematics that he’s railing against the kind that the layman would be educated with ? Or is he talking about a type that is left to math majors and such ?
No Western mathematician that I’m familiar with (at least any who are not kooks) ever made this claim. Indeed, the fact that one can arrive at a false conclusion when starting from invalid axioms is fundamental to proof theory, and is one of the first things taught in any kind of formal logic course, as well as informal philosophical argumentation courses.
No one claimed that, either.
I think anyone who would make this claim (or wrongly ascribe the claim to a group of people) does not understand what deductive and empirical proof are. One can not be superior to the other, because they are used for entirely different things. Maybe Raju is arguing that traditional mathematics says that deductive reasoning can accomplish things which empirical observation can not. That’s true. So is the inverse. They are different tools for different jobs.
What’s “valid knowledge?” Why is knowledge gained from axiomatic, non-empirical systems less valid than other kinds of knowledge?
I think that the way we teach mathematics to kids is badly out of date and curriculums should be redesigned. I don’t think C. K. Raju would be my top choice for doing so.
At the college and graduate school level, there’s plenty of reason why math and science students should study philosophy of mathematics. I was required to take a course in “Math and Society” junior year, and I can still say it was one of my most important college courses, even though the professor was slightly crazy. Actual mathematicians should know about the deductive vs empirical, intuitionist vs rationalist controversies, Zorn’s Lemma, the Banach-Tarski paradox, and things like that. But it would be pointless to try pushing those topics into high school, much less elementary school.
> For example, the very first proposition of “Euclid’s” Elements has an invalid deductive
> proof. But for 8 centuries that book was mistakenly regarded by all the foremost minds
> in the West as the model of deductive proof, when, in fact, there isn’t a single valid
> deductive proof in it, as Bertrand Russell too emphasized.
Let’s see a demonstration that there isn’t a single deductive proof in the Elements.
Throughout the history of mathematics, deductive proof and the axiomatic method on the one hand, and empirical verification and real-world application on the other hand, have been complementary.
I assume this is a reference to the fact that some of Euclid’s proofs (including the first one) rest on assumptions that can’t be justified from his axioms. From the Wikipedia article on Euclidean geometry:
Wasn’t there a thread some time ago about some academic activist claiming that science and math courses discriminated against women because the answers had to be exact or something?
I was thinking more of the Alan Sokal article. Good parody should be hard to distinguish from sincerity. I am pretty sure Mr. Raju is serious, but not 100%.