MEDIA
“Pure” Mathematics: The Good, the Bad and the Ugly
Mathematics from a normative and communications perspective.
Not so long ago I completed a poll among ten members of our academic community, five from our Mathematics Department and five from the outside. The latter included college counselors and a dean from one of our colleges. The poll consisted of having them answer the following question:
“What is the first thing that comes to mind when you assess ‘pure’ mathematics?”
The result produced the following off-the-cuff answers:
- “it has no motivation in applications”
2. “it has no motivation in applications, even though I understand it is beautiful ”
3. (i) “lack of motivation;”, (ii) ”Who cares!”
4. (i) “lack of applications”; (ii) ”lack of people who care”
5. “don’t talk to the general public [about “pure” mathematics]”
6. “not useful”
7. “no practicality, no usefulness, no bearing on anything”
8. “beautiful, but does not care about applications”.
9. “implies mathematics is contaminated”
10. “is outside the context of the real world”
The criticism implied by these answers is summarized bluntly by # 7: “pure” mathematics has “no practicality, no usefulness, no bearing on anything”. Conversely, mathematics which is practical and useful is viewed as “contaminated”, i.e. “impure”, because it is about the world.
What the ten answers highlight is a normative error originated by Plato and promoted by seventeenth and eighteenth century philosophers. They claimed that there is a fundamental cleavage in knowledge, including quantitative knowledge; it divides mathematics¹ into two mutually exclusive (and jointly exhaustive) types, “pure” and “impure”.
Many, if not most, mathematicians by and large do not worry about the chaotic and dire cultural consequences of epistemic errors such as “pure” mathematics. They unwittingly embrace this error by simply equating “pure” mathematics to being abstract and logically rigorous and then by committing the other part of the same Platonic error: denigrating applications.
However, both logical rigor and applications are crucial. Without the first, we cannot be certain that our statements are true; without the second, it does not matter whether or not they are true.
The pure-impure dichotomy drives a wedge between the two. It is a breach between (a) the physical world (reality) and (b) conceptual and mathematical statements about it. This dichotomy presents the following Hobson’s choice: conceptual statements in terms of rigorous and non-trivial mathematics are “pure” and hence are disconnected from the world and morally good, while physically concrete observations about it that are formulated in terms of mathematical statements are “impure” and hence inferior or mathematically trivial at best.
“Pure” mathematics is one of the symptoms of a philosophic auto-immune contagion, known in technical philosophy as the analytic-synthetic dichotomy², which is its DNA. It is a dogma according to which a “necessarily” true proposition cannot be factual, while a factual proposition cannot be “necessary”. This contagion latches itself onto the very base of a person’s mind when he embraces it. Indeed, it is an important historical fact that this dogma, which is devastating for science and mathematics³, has been at the root of the collapse of philosophy (during the past two centuries) into rank irrationalism⁴, a belief that reality and scientific theories are “social constructs” based on peer pressure, and that reason is not valid or useful.
The “pure” mathematics terminology⁵, with its implied pure-impure dichotomy, serves only to condition the mind into its acceptance, and should be abandoned. What is required in this context is the appreciation and the acquisition of the appropriate method of thinking, the means of forming concepts from concretes, of inductive reasoning⁶, the means of generalizing from the particulars to abstract principles.
Aside from substituting “theoretical” for “pure” in referring to abstract and rigorous mathematics, a more ambitious approach, and more effective in the long run, would be one which is more general: deplatonize⁷ mathematics to immunize it against the DNA of this philosophic contagion.
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¹ and in some circles, even physics
² The Analytic-Synthetic Dichotomy By Leonard Peikoff
³ Page 75–77 in Explaining Postmodernism by Stephen R.C. Hicks.
⁵ “Theoretical Mathematics” would be a superior terminology. This is discussed in the “Preface ” of the textbook Linear Mathematics in Infinite Dimensions
⁶ THE LOGICAL LEAP: Induction in Physics by David Harriman
⁷ i.e. purge Plato’s mystical elements (such as “see with the mind’s eye”, “intuition”, ESP, “revelation”, …) from one’s basis of arriving at knowledge.
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Ulrich Gerlach, Department of Mathematics, Ohio State University.
