Defining Mathematics

New Yorker magazine essay tries to define mathematics:

Mathematicians know what mathematics is but have difficulty saying it. I have heard: Mathematics is the craft of creating new knowledge from old, using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can’t see, whose presence is confined to the imagination. A proto-text whose existence is only postulated. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition. The thing that scientific ideas, as they grow toward perfection, become. An ideal reality. A story that has been written for thousands of years, is always being added to, and might never be finished. The largest coherent artifact that’s been built by civilization. Only a formal game. What mathematicians do, the way musicians do music.

Those are good, but this gets to the heart of the matter:

Mathematicians live within a world that is essentially certain. The rest of us, even other scientists, live within one where what represents certainty is so-far-as-we-can-tell-this-result-occurs-almost-all-of-the-time. Because of mathematics’ insistence on proof, it can tell us, within the range of what it knows, what happens time after time.

I will have to ponder this:

In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake.

Denyse O'Leary writes:

The Oregon Department of Education (ODE) recently encouraged teachers to register for training that encourages “ethnomathematics,” an education trend that argues, “among other things, that White supremacy manifests itself in the focus on finding the right answer”:

“The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so,” the document for the “Equitable Math” toolkit reads. “Upholding the idea that there are always right and wrong answers perpetuate objectivity as well as fear of open conflict.” …

An associated “Dismantling Racism” workbook, linked within the toolkit, similarly identifies “objectivity” — described as “the belief that there is such a thing as being objective or ‘neutral'” — as a characteristic of White supremacy.

Instead of focusing on one right answer, the toolkit encourages teachers to “come up with at least two answers that might solve this problem.”

Sam Dorman, “Oregon promotes teacher program that seeks to undo ‘racism in mathematics’” at Fox News (February 11, 2021)

Opponents of the new trend offer varying accounts of its origin — perhaps it results from in changes in overall philosophy of life or perhaps from the practical need to placate teachers’ unions, which may have various objectives apart from enabling numeracy in students. No matter, not only are x and y under attack but so is 2 + 2 = 4.

It is embarrassing how many respected scientists allow this nonsense to persist without saying anything. I am happy to see that physicist Lawrence Krauss wrote a Quillette essay denouncing some of the leftist propaganda:

Social justice activists have been arguing for some time that scientific societies and institutions need to address systemic sexism and racism in STEM disciplines. However, their rationale is often anything but scientific. For example, whenever percentages in faculty positions, test scores, or grant recipients in various disciplines do not match percentages of national average populations, racism or sexism is generally said to be the cause. This is in spite of the fact that no explicit examples of racism or sexism generally accompany the statistics. Correlation, after all, is not causation. Without some underlying mechanism or independent evidence to explain a correlation of observed outcomes with population statistics, inferring racism or sexism in academia as the cause is inappropriate.

One might have hoped for more rigor from the leadership of scientific societies and research institutions. Alas, this has not been the case.

My guess is that other scientists are afraid to say this out of fear that they will be canceled. Krauss only dares because he has already been canceled. His enemies tried to get him fired, based on gossip. I don't know if they succeeded or not.