6 ms·

Numbers Algebra Geometry Probability Analysis Dynamics I loved this talk, but these concepts are like an attempt at dimensional reduction of math research

by iTokio 16d ago

Numbers Algebra Geometry Probability Analysis Dynamics I loved this talk, but these concepts are like an attempt at dimensional reduction of math research, science, the academics knowledge. I would have loved to have his thoughts on the mathematical mind, the process, how to reason, infer vs deduct, abstract, prove.. I don’t really know, what are the primitives, essential concepts of math reasoning?

Isamu 16d ago
So… numbers, algebra, geometry, probability, analysis, dynamics are not the primitives or essentials ?
- MrbroJangles 16d ago
  Imo it's not numbers at all but linked to our awareness of physical relationships Making it about numbers is like making it about cans when it's more about grasping adding one can to a bag of cans, adding 100 cans (multiplication), or the inverse with subtraction and division Which is why I never liked numbers before algebra which then chucks numbers in the bin more or less. Numbers are just syntax meant to represent $anything; 1.5 can be half a pill and a whole pill or T or A; numbers are euphemism. That they can be infinitely big and yadda yadda isn't that meaningful in and of itself and that little bigness is all due to additive qualities of physical space Geometry is addition or subtraction of shape It's all built on 4 operations we see in daily life all the time
  - andsoitis 16d ago
    > Numbers are just syntax Numerals are syntax; numbers are mathematical objects. “5”, “V”, “101₂”, and “|||||” are different representations of the same number. A variable such as x is closer to what the statement means by something that can stand for arbitrary things. > It’s all built on four operations Elementary arithmetic emphasizes +,-,×,÷, but mathematics isn’t reducible to them. Mathematics studies operations and relations such as composition, exponentiation, differentiation, integration, limits, logical implication, set membership, mappings, probability, topology, symmetry, transformations, equivalence relations, and many others.
- deleted 16d ago
  [deleted]
binyu 16d ago
Why do you feel this is a reduction? If anything, it is an attempt to summarize the various areas of math and how they relate to each other.
- alansaber 16d ago
  Isn't every summarization a reduction?
  - andsoitis 16d ago
    > Isn't every summarization a reduction? Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.
- iTokio 16d ago
  It’s like principal component analysis, the main axis, I don’t mean reduction in a negative way.
SMEbooop 16d ago
[dead]
bbeonx 16d ago
these might not be 100% complete, but i think this does a reaaalllly good job at capturing the vast majority of mathematics. i'm sure you could come up with another breakdown, but these also have the benefit of tracking roughly with history. numbers and geometry (euclid), then eventually algebra. probability was a fundamentally new way of looking at the world. analysis comes via newton/leibniz and then dynamics tries to tackle complex systems (kinda where newton left off, e.g., 3-body problem type stuff). also: dimension reduction is not a bad thing. this is like a decomposition: we can decompose so much of what research mathematics is doing into 6 different basis elements. that's pretty darn neat. (edit: liebnitz -> leibniz)
kian 16d ago
Read Polya's "How to Solve It" :)
rramadass 16d ago
Related: From a previous HN discussion of Terence Tao's Six Math essentials book (to be published) my comments pointing to a similar book by John Stillwell (covers Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic) - https://news.ycombinator.com/item?id=47116399 https://news.ycombinator.com/item?id=47116399
RossBencina 16d ago
The essential concepts of mathematical reasoning, if there are such things, are the concern of mathematical logicians, or maybe even psychologists, not, in general, of working mathematicians. One of my professors once told me something to the effect of "if you think you are going to learn any of that here, you are in the wrong place." If you are interested in Terence Tao's personal mathematical inner world, he touches on that during his interview with Lex Friedman, which I think you might find interesting. Interviews with Kevin Buzzard sometimes touch on these themes too.
- iTokio 16d ago
  Thank you, I will watch these interviews!
  - hooda 11d ago
    If you want to follow Terence Tao's new interviews, podcasts, blog posts, etc. and get an email delivered promptly to your inbox, you can do it here: https://recordal.app/people/terence-tao https://recordal.app/people/terence-tao Disclaimer: I built this tool a couple of weeks back.
- winwang 15d ago
  "maybe even psychologists" -- unironically true, but maybe I am talking about something different from you. The basis of (higher-level) math seem to mostly be "am I psychologically (emotionally...?) comfortable with accepting annoying ideas?" At least from my experience.
killerstorm 15d ago
This video actually explains math reasoning on a high level. Have you actually watched it?

by iTokio 16d ago

Isamu 16d ago

So… numbers, algebra, geometry, probability, analysis, dynamics are not the primitives or essentials ?

MrbroJangles 16d ago
Imo it's not numbers at all but linked to our awareness of physical relationships Making it about numbers is like making it about cans when it's more about grasping adding one can to a bag of cans, adding 100 cans (multiplication), or the inverse with subtraction and division Which is why I never liked numbers before algebra which then chucks numbers in the bin more or less. Numbers are just syntax meant to represent $anything; 1.5 can be half a pill and a whole pill or T or A; numbers are euphemism. That they can be infinitely big and yadda yadda isn't that meaningful in and of itself and that little bigness is all due to additive qualities of physical space Geometry is addition or subtraction of shape It's all built on 4 operations we see in daily life all the time
- andsoitis 16d ago
  > Numbers are just syntax Numerals are syntax; numbers are mathematical objects. “5”, “V”, “101₂”, and “|||||” are different representations of the same number. A variable such as x is closer to what the statement means by something that can stand for arbitrary things. > It’s all built on four operations Elementary arithmetic emphasizes +,-,×,÷, but mathematics isn’t reducible to them. Mathematics studies operations and relations such as composition, exponentiation, differentiation, integration, limits, logical implication, set membership, mappings, probability, topology, symmetry, transformations, equivalence relations, and many others.
deleted 16d ago
[deleted]

binyu 16d ago

Why do you feel this is a reduction? If anything, it is an attempt to summarize the various areas of math and how they relate to each other.

alansaber 16d ago
Isn't every summarization a reduction?
- andsoitis 16d ago
  > Isn't every summarization a reduction? Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.
iTokio 16d ago
It’s like principal component analysis, the main axis, I don’t mean reduction in a negative way.

SMEbooop 16d ago

[dead]

bbeonx 16d ago

these might not be 100% complete, but i think this does a reaaalllly good job at capturing the vast majority of mathematics. i'm sure you could come up with another breakdown, but these also have the benefit of tracking roughly with history. numbers and geometry (euclid), then eventually algebra. probability was a fundamentally new way of looking at the world. analysis comes via newton/leibniz and then dynamics tries to tackle complex systems (kinda where newton left off, e.g., 3-body problem type stuff). also: dimension reduction is not a bad thing. this is like a decomposition: we can decompose so much of what research mathematics is doing into 6 different basis elements. that's pretty darn neat. (edit: liebnitz -> leibniz)

kian 16d ago

Read Polya's "How to Solve It" :)

rramadass 16d ago

Related: From a previous HN discussion of Terence Tao's Six Math essentials book (to be published) my comments pointing to a similar book by John Stillwell (covers Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic) - https://news.ycombinator.com/item?id=47116399 https://news.ycombinator.com/item?id=47116399

RossBencina 16d ago

The essential concepts of mathematical reasoning, if there are such things, are the concern of mathematical logicians, or maybe even psychologists, not, in general, of working mathematicians. One of my professors once told me something to the effect of "if you think you are going to learn any of that here, you are in the wrong place." If you are interested in Terence Tao's personal mathematical inner world, he touches on that during his interview with Lex Friedman, which I think you might find interesting. Interviews with Kevin Buzzard sometimes touch on these themes too.

iTokio 16d ago
Thank you, I will watch these interviews!
- hooda 11d ago
  If you want to follow Terence Tao's new interviews, podcasts, blog posts, etc. and get an email delivered promptly to your inbox, you can do it here: https://recordal.app/people/terence-tao https://recordal.app/people/terence-tao Disclaimer: I built this tool a couple of weeks back.
winwang 15d ago
"maybe even psychologists" -- unironically true, but maybe I am talking about something different from you. The basis of (higher-level) math seem to mostly be "am I psychologically (emotionally...?) comfortable with accepting annoying ideas?" At least from my experience.

killerstorm 15d ago

This video actually explains math reasoning on a high level. Have you actually watched it?