Can one explain schemes to biologists? (2014)
dam.brown.edu
dam.brown.edu
* among the general public people feel it is acceptable to admit lack of basic skill or understanding in mathematics (at times even arithmetics) whereas they would be ashamed to admit it with respect to reading or writing,
* popular science books are often written without a single formula (and according to a preface in one popular science book editors and publishers generally reject a book that tries to sneak some in, their apparent justification being that each formula halves the sales),
* and it turns out that even STEM scientists, in this case biologists, appear to have gaps in their understanding of basic mathematics like analysis and algebra.
I'm very curious as to what might be behind this isolation of mathematics. Bad teachers? Mathematicians' own disregard for applications? Some sort of self-perpetuating myth about inaccessibility of maths for the common person?
When I teach courses that I have no control over, I see over and over again tricky questions that have no relevance to actual mathematical understanding or use. At the same time, the majesty, wonder, and beauty of mathematics is expressly avoided as being "too much for students to handle".
In essence, we treat students as if they are computers learning rules to apply. But mathematics is about artistry and creativity. Being cold and harsh, being factory like, deprives one of the part of the brain necessary to do real mathematics.
Adults routinely tell me "I did well in mathematics until...and then I stopped understanding. I stopped enjoying, I struggled with my grades, and I can't do math."
Bottom line, people get labeled as bad at mathematics and they accept it. And it is only getting worse with common core tests.
We might also even see future generation being bad at reading, etc. Already I hear first graders saying "I'm not a reader because I do poorly on the reading tests."
It is sad because mathematics is one of the great gifts to humanity. It is breathtaking when used and done properly.
Innumeracy is a very similar problem to illiteracy, it has a broad impact on your ability to learn and reason about the world - but it is much more socially accepted and (hence?) much more prevalent.
It was previously discussed here: https://news.ycombinator.com/item?id=8819811
Hilbert said "Wir müssen wissen — wir werden wissen!", and just the day prior Gödel demonstrated the futility of axiomatic set theory. Yet, still today, ZFC set theory is assumed unless it is stated otherwise.
There are Luddites among mathematcians, and Turing died for their sins.
ZFC admits the first by not limiting the range of the bound variables of the axiom schemas of Separation or Replacement (unlike the Separation axiom schema of Bounded Zermelo Set Theory), but it disallows the second (due, for example, to Foundation).
So at least one form of self-reference, impredicative definitions, are permitted by ZFC.
ED: Thanks for the reference. It appears to be a much more accurate account of the rift which I tried to talk about. I have previously been informed that my conclusion that ZFC does not permit recursion, as in a cyclic graph, was correct, and this certainly is further evidence regarding this issue.
I don't think the issue is mathematics communication. The math community is pretty obsessed with spreading knowledge, and some fields (e.g. econ) make fantastic use of math as a communications tool. The issue is rather that a lot of folks simply refuse to learn it and complain when others use it.
Either way, I can speak in a strict fashion which sort of jives with formal logic. I referenced Hilbert's program and the assumption that mathematics has a foundation: https://en.wikipedia.org/wiki/Foundations_of_mathematics#Fou...
I make a leap of faith here; that humans can not abandon the need to find meaning. From this I surmise that there does exists a deep rift in the mathematical community. - Both Formalism and Intuitionism remain today, and if an understanding has been reached between these schools then I must have become impervious of the fact...
I assume you are referring to the axiom of foundation which implies, for example, that you cannot find three sets A, B and C such that A ∈ B, B ∈ C, C ∈ A. This is not an obstacle to talking about self-reference, recursion or cyclic graphs, etc. The axiom of foundation keeps the membership relation from having cycles, but it doesn't keep you from talking about other relations that might have cycles! Indeed, defining functions recursively is completely standard practice in mathematics, and is formalizable in ZFC with no problem. Also, the most common definition of graph is explicitly in terms of set (a graph is an ordered pair (V,E) where E is a set whose elements are pairs of elements of V)and easily formalizable in ZFC. That you can't have ∈-cycles never comes up in, say, the study of cycles in graphs or of recursively defined functions.
I can't see how your implication works either, can you make it explicit?
What I am trying to say is that you do not need mathematical formulas to explain mathematical concepts, but certain vocal academics have managed to push a view that math is nothing but formulas AND that there is no controversy here. There is, in fact, a lot of controversy and this does lead back to Grothendieck since his work does not support the formulas-only view.
This isn't really a thing, which may explain why I'm having trouble understanding what you are trying to say. To be more concrete: from what I can understand of your somewhat incoherent characterization it does not match any mathematical community I know of - it may exist, but if so it is not mainstream.
Further, differences in communication rarely stem from foundational issues as much as merely the social construct that practitioners in separate fields have less to say to one another than those in the same field. This leads to semantic drift and development of independent metaphor technologies. Translation necessarily becomes more expensive and so higher ROI is required in order to motivate it.
"Practical" pure math appears to be based on formalism rather than intuitionism. Because foundations and therefore intuition does not hold formalism back, these practitioners drift further away from being able to communicate their metaphors to whose who rely on intuition; such as biologists.
- And this is my opinion.
What I am trying to say is that math is a discipline of philosophy, that the integers are divine, and that the straightedge and compass ought to be enough for anybody. ;) I often rely on my intuition for abstract thought, and repeatedly find that others have reached the same conclusions that I do. Therefore (my own) intuition is repeatable, and therefore intuition is scientific.
Juxtapose formalism, which seems to say that math is solely the practice of inventing rules and following them to the end. - I understand why this idea has its own beauty, haughty as it might seem, but at the same time it is a very strange and in my opinion frightening kind of beauty because it intends to remain unknowable.
Turing, who I have apparently proclaimed a saint, seems to observe that both of these paradigms are needed for some kind of universal, and altogether anthropomorphic, program. I would be hard pressed to dismiss any idea which appears beautiful...
It appears to me that in software engineering the situation is a lot better. Anecdotes from my own recent experiences:
- polynomials (encountered when reading about CRC codes and elliptic curve cryptography),
- infinitesimals and vectors (encountered when recalling Newton's third law when working on a small Android game using libGDX and box2d),
- complex numbers (encountered when learning Go and reading about AC circuits).
I haven't encountered complex space, but my guess is that it's similar to the familiar real vector space but permits complex numbers as vector components and coefficients in linear combinations.