Explaining Mathematics

[daylen]

Okay, this is not a competition, but I thought it might be fun.

To participate either ask a question or describe a particular item (or add your own entry). It would probably be best to have a mixture of one-sentence answers, one-paragraph answers, and so forth to capture different complexity levels with different analogies. I tried to keep the list short and simple. This was much harder than I thought it was going to be initially. Math is huge.

1. Complex number
2. Prime number
3. Induction
4. Modulo
5. Derivative
6. Integral
7. Differential equation
8. Vector space
9. Hilbert space
10. Topological space
11. Metric space
12. Group
13. Ring
14. Graph
15. Category

Bonus points
16. Cardinal number
17. Determinant
18. Distribution (not the statistical kind )
19. Hamiltonian
20. The Spectrum

[daylen]

I'll try very short one-sentence answers. I used 'The Princeton Companion to Mathematics' for later entries but tried to simplify and rephrase slightly.

1. A real and imaginary number added together.
2. A number with no divisors besides one and itself.
3. A type of proof that starts with a base case, establishes the incremental case, then induces all cases.
4. Remainder after division.
5. Slope of tangent line on a curve at a point.
6. Area under curve.
7. Equation containing information about how variables change with respect to each other.
8. All the combinations of a set of vectors that can be added together or multiplied by a real number.
9. Vector space with added constraints to preserve angle.
10. Has just enough structure for continuity to exist.
11. Topological space where distances exist.
12. Set of elements where any two can be combined into a third still in the set.
13. Set of elements closed under addition and multiplication.
14. Set of vertices along with the set of edges between them.
15. A mathematical structure along with a structure preserving map.

16. Measure of set size using bijection.
17. Scalar computed from a matrix with a simple yet cumbersome procedure.
18. Generalized notion of a function that deals with objects as opposed to points.
19. An object that largely determines how the state of a physical system evolves across time.
20. Generalization of eigenvalues on matrices to linear operators.

[daylen]

This will probably be a monologue, but from time to time I go through a math phase so I will keep this thread in mind.

[daylen]

Mathematics is the art of making the strongest conclusions with the fewest assumptions. Someone said something like this once.

[daylen]

Ha, I wrote multiples instead of divisors on prime number entry. Silly me.

[daylen]

I will talk some philosophy of mathematics here too.

So one way to think of math is as a constraint hierarchy where each level has objects that obey those constraints. Each level can also be associated to theorems or conclusions that can be drawn for such objects.

When starting out in mathematics you learn by bridging the gap between how reality works with one of these levels. Typically, this requires making many constraints so that this intuitive step is easy to leap(*). Part of advancing in mathematics is loosening these constraints while still attempting to retain intuition.

The ability to manipulate mathematical expressions is completely different from the ability to intuit what a mathematical expression represents in reality. This is where the gap between Ti and Ni can be greater than anywhere else. Beyond around four or five dimensions, math seems to be almost entirely Ti with small Ni leaps that become more like Ne guesses. Dimensions can be interpreted in many ways, so this is just a rule of thumb.

Physics is also super important in developing mathematical intuition. Many physicists probably understand math better than many mathematicians in the sense that they can intuit what a function looks like or how it behaves under transformations. Historically, physics has influenced math more than the reverse. Perhaps this is not as true these days, but pure math and theoretical physics research now is so far up in the clouds that they basically form a religion. Not to say there isn't exciting research being done with the merger between computation, math, physics, economics, and so forth.

Applying CCCCCC to math/physics is a bit tricky since you basically need to jump to computing immediately or else you are not really 'doing' anything. Mathematical coordination is extremely valuable in industry because of algorithms, big data, yadda yadda... This is another aspect where physicists tend to shine because of the way their training is coupled to real world data. There seems to be many mathematicians that can compute very well but never understand why they are doing it. These are probably also the same people that did very well in high school and college, then were encouraged or rather enslaved into a graduate school doctrine working as mediocre computers that speak English.

(*) Many very smart people struggle with early math precisely for this reason. They feel as if there are assumptions being made without adequate explanation (e.g. Jung).

[horsewoman]

(*) Many very smart people struggle with early math precisely for this reason. They feel as if there are assumptions being made without adequate explanation (e.g. Jung).

This is interesting. My kid scored pretty high in maths in the IQ test they do with gifted children, but primary and middle school maths is a book with seven seals to her. I cling to the hope that this particular knot will open once things get more complex...

[daylen]

This is not uncommon. IQ tests often require continuing a sequence, filling a gap in a sequence, and filling a gap in a 2D grid. This form of reasoning is mostly associated with Ni and Ti which are not all that useful in a school setting (even at the university level to some extent). Often gifted kids will especially hate multi-step processes where there is a clear short-cut to the answer.

One example is long division, which is something I refused to learn because it was tedious, and I could not see any reason why I needed to know it. Never needed it for my math degree, then I had to learn it when teaching high school. School is build around Te-Si, so it is no surprise that many kids who can do well on IQ tests fail to meet expectations.

[jennypenny]

This is interesting. My kid scored pretty high in maths in the IQ test they do with gifted children, but primary and middle school maths is a book with seven seals to her. I cling to the hope that this particular knot will open once things get more complex...

DS was the same but when he took geometry in high school it was like he went from seeing math in black and white to seeing it in color. Now he's considering computational math as a second major/minor in college. Keep the faith.

[jacob]

The ability to manipulate mathematical expressions is completely different from the ability to intuit what a mathematical expression represents in reality. This is where the gap between Ti and Ni can be greater than anywhere else. Beyond around four or five dimensions, math seems to be almost entirely Ti with small Ni leaps that become more like Ne guesses. Dimensions can be interpreted in many ways, so this is just a rule of thumb.

People working with high-math are INTJ and INTPs. I've never met an exception to this rule. (ISTJs drop out and become middle managers. ENTPs join the student party committee. ENTJs switch to econ.) This provides an interesting and sometimes infuriating contrast between Ni->Te (INTJ) and Ti->Ne (INTP).

Iff Ni (INTJ) groks the math, solutions can be correctly intuited. This often seems like magic to INTPs. However, communicating this intuition via T (Te) demands a lot of mental energy, because from the Ni perspective only the origin and destinations are known. Therefore a "proof" requires the engagement of the auxiliary Te to try to connect the origin and destination w/o Ne assistance to suggest the next step. The process thus becomes mechanical and progressed from either end until it either fails or meets in the middle.

Ni is not a good dominant function in terms of learning math as it's practiced at the undergraduate level. It's pretty good at suggesting useful perspectives to mathematicians at the professional level. This is why physics often leads math suggesting new directions, etc.

Mathematicians speak of maturity as they get more experienced. This may have to do with developing Ne... knowing how to guess to go in a productive direction. Physics maturity (Ni) has to do with boiling a problem down to its essentials, thus reducing the needed load on the Te.

[daylen]

@Jacob I have heard people type Bertrand Russell as an ENTP, but I have not really looked too deep into this. From reading some of his work, both INTP and ENTP seem plausible. I definitely agree that types other than INT are rare.

I can also see the development of Ne (coupled with Si) as being related to mathematical maturity. Much of this I think has to do with keeping track of bits of information along a calculation/proof tail, sort of speak. If the bits do not match the strength of the unraveling conclusion then you can smell something fishy going on. Another way this may come out is when you feel the conclusions should be stronger given all the information you have, or even another way would be detecting informational overlap. This can all be done in the mist of a calculation/proof indicating an Ne-Si flavored maturity.

[7Wannabe5]

People working with high-math are INTJ and INTPs. I've never met an exception to this rule. (ISTJs drop out and become middle managers. ENTPs join the student party committee.

lol- True enough, but since ENTPs also love a challenge, I would like to know what qualifies as "working with high-math?" I only got one wrong on math section of GRE when my brain was much younger, and I recently got 4 stars on all 4 sections of the exam qualifying to teach secondary level math. Generally, I like abstract/proven and dislike applied/approximation. However, it is on my bucket list to learn the mathematics necessary for dynamic systems analysis, but not for "work" purposes. I want to do this for the same reason I felt compelled to fill my dollhouse toilet with real toilet water when I was a child.

[jacob]

I consider "high-math" to be of the "theorem-proof" variety such as what math-majors would see and focus on. I distinguish this from what I'd refer to as "calculation" which is the applied-use version of math however complicated such as what would be introduced to STE-majors on a need-to-know basis and HS graduates on a teach-to-the-test basis. English only has one word covering both concepts but other languages have distinct terms delineating the difference.

Add: Math = why, calculation = how.

[daylen]

@jacob I know you have talked about Ni quite a bit, but I am still utterly fascinated by it.

You have talked about the latticework, constellations, and constellations of constellations, but I am curious how all of this feels and what entering/exiting is like? Is there any apparent topology to it? ..like are you aware of the curvature (flat, positive, negative)? ..does it actually feel like it could map onto the DMN or part of it (like a shell or rather a rough shell)? Is there a quick access mode that goes deep? .. a scanning mode that stays at surface level?

Also, in a different direction related to this thread.. about how many nodes do you have to deal with PDE's? In other words, how do you store the essence of PDE's in the latticework? I know you have to Te an answer, but just a vague description would still be interesting for me. The way it works for me is very categorical and therefore hard to master given the variability and complexity of PDE's. Like, for me there are some basic distinctions (homogeneity, elliptic, hyperbolic, parabolic, etc.) and some particular examples (wave, heat, etc.), but I have a very hard time understanding the essence or behavior of a PDE just by looking at it.

[fiby41]

21 Number systems

The precursor to the decimal place value system is the Bhūtasaṃkhyā system where zero is denoted by any of words for space, one is represented by moon as our planet has only one, two by eyes, three by the modes of material nature, four by the number of Vedas, five by an arrow,...
Application: provides flexibility when composing verses when you have to juggle a fixed number of syllables in a meter while maintaining the rhyme scheme.

Āryabhaṭa's system uses

No need to memorise arcane formulae that make your eyes roll over just chant makhi bhakhi phakhi dhakhi... (it rhymes!) and you have the sine series expressed as sum of successive angles.

There are also the Kaṭapayādi and Akṣarapallī.

Mayans did it in multiples of one, two and five only.
Mesopotamians had a base 60 number system.
Binary, octal, hexadecimal most are already familiar with.
You can count in hex on one hand using the thumb as pointer.

[daylen]

@jacob I should also note that I can visualize the behavior of PDE's well. Like I can see how the heat or wave equations work in 1D, 2D, and 3D over time, but I cannot easily link this to the equations. I suppose this is the same problem you talked about earlier where physicists can intuit the math iff they know the math. So, I suppose this is just a problem of me not knowing the math well enough. When I took a PDE course, I was not that interested at the time since my physical intuition was lagging, so I sort of slacked. I imagine that you have made this link much better though, hence why I am curious as to what it looks like. Yet, this was probably a lot to ask.

[Tyler9000]

DS was the same but when he took geometry in high school it was like he went from seeing math in black and white to seeing it in color. Now he's considering computational math as a second major/minor in college. Keep the faith.

Same for me. I have an engineering degree with a math minor, but TBH I never really grokked most math that I couldn't visualize. Sure, I was good at rote memorization of equations to pass a test, but that doesn't automatically mean you really understand what you're learning. But give me geometry, vectors, or even differential equations that I can visualize as motion, and it's a piece of cake. Some people just think differently, and math takes off when you find your strength.

BTW, does someone have a good reference that explains Ti / Ni? I found this useful, but I assume you guys may have more. I get the general idea, but I feel like there's a lot of abstract Ni information in this conversation that INTPs like me don't fully understand.

[daylen]

@Tyler9000 Jacob and I have discussed Ti / Ni from time to time. Not sure how much digging you want to do. I am working on a site currently to serve as a reference for this sort of thing. As I update it I will make announcements in my journal.

The way you describe your experience with math seems like typical INTJ struggles, but this happens to a lot of types even if they use Ti. Both Ni and Ti can be abstract in their own ways.

[Tyler9000]

Thanks for the tip. Apparently I haven’t been reading enough ERE lately and have some catching up to do.

[7Wannabe5]

Add: Math = why, calculation = how.

Gotcha. I like to know "why" so I can derive "how", but I recognize that it's not always possible or practical. Anyways, I am an eNTP, and I am VERY aware that I am in my introverted function when I am doing math. I feel cold when I attempt to do it for too long.

22. Function: 20th century economist's wet dream.