- There is nobody that can fill in that sentence. Perhaps for themselves, but it would not be valid for everyone. Therefore I will not. Rather, I will answer the question with the complexity it deserves.
0.99999... = 1 (Infinity)
Let me invite you on a tour to the Mountain of Mathematics, where we will find three towers. On our map, you see that the three towers are named:
- Pure mathematics
- Unreasonably effective mathematics
- Mathematical Modelling and Statistics
Thing is, at first they all wanted their tower to be called either "true mathematics" or "interesting mathematics", but that never sat well with the others. You will still hear them whisper those names between friends. On this tour, I will be your guide, as I have a room in each of these towers, rooms in which I have lived at least two years. (I have done university level mathematics of all three types, for at least 2 years, each.)
On the way up the Mountain, our group talks about what they think advanced mathematics is.
- "It is, of course, multiplying large numbers! And very good professors can do it in their heads" says one, you suspect he did not learn much mathematics at all.
- "It is, of course, solving very hard equations for x! And very good professors can deal with x to the power of four and logarithms and such" says another, and you suspect she did not have mathematics in high school.
- "It is, of course, differentiating and integrating difficult functions, and to solve differential equations! And very good professors can solve anything." says a third one, and you suspect he did have mathematics in high school. But he has not studied at the university.
It is inherently hard to guess what the next level of mathematics is, how it looks, how it feels, because if you did, that is where you would be. (1) But let us try, anyway.
Climbing the Mountain
- Brain teaser: Why does this work? Compound interest: 1.03^27 is approximately 0.81*2.7
- First there is a level of basic knowledge… Knowing the symbols, what a proof is, how to compute something, familiarity with functions, solving simple equations for unknowns, etc. ("Basic" at this level of mathematical knowledge!)
- Underlining the answers because it is absolutely correct. (Is that true, is it absolutely correct? And when there is no single answer? A Proof, a logical argument?)
- Hadarar nilte 15 ankrets, elfi sura 3 univ, quelta iva ana est? (Social intelligence is not mathematics.)
- Those who try to climb this mountain are not average, but very intelligent! (But, who do they compare themselves to?)
- Poems vs Mathematical text (Norwegian links)
- Lonely genius or collective genius?
- The Towers have bridges between them, but some look very fragile.
There are other ways to subdivide mathematics, the most common is by subject. The subdivisions here are in line with how people think about their own mathematics, and how it feels to do mathematics, it is divided according to interests and human behaviour.
The Tower of Pure Mathematics and Logic
Mathematics as its own reward. (Algebra, analysis, topology. Proofs.)
- BBC - Horizon - 2010 - To Infinity and Beyond (Infinity, see 0-01:30 & 14:11-19:10)
- Google: hilbert infinite hotel
- https://en.wikipedia.org/wiki/Fractal_dimension (Fractal)
- Multiplication modulo prime
- The most natural definition of equally many: Fractions countable (Image), Reals uncountable (Image)
- https://en.wikipedia.org/wiki/Quintic_function#Finding_roots_of_a_quintic_equation
- Non-measurable_set
- Gambit: Reductio ad absurdum
- Turing (Computer Bombe)
- Gödel's testimony
- Cryptography: RSA algorithm
- Even 1+1=2 can be made complicated
- The room where I lived, Algebra, Discrete math, Analysis, Ring theory
The Tower of Unreasonably Effective Mathematics
Mathematics as the Language of Science. (Vectors, differential equations, discretizations and numerics, Nabla/Delta.)
- https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
- When you write something down in the language of mathematics, you find out what you really mean
- You can test any equivalent statement in experiments
- The models are not approximations, they seem to fit perfectly
- https://www.youtube.com/watch?v=1TKSfAkWWN0 (Special relativity and Magnets)
- Newton
- Law 2 and 3: complicated geometry -> simple vector math
- W = F*s, I had to invent integration (Apple falling with air resistance, Energy conserved)
- Noether
- Noether's theorem (Conserved energy, momentum, angular momentum)
- Einstein
- Developing useful tools
- Natural numbers, Negative, Rational, Irrational, Imaginary
- What is a vector? Can a function be a vector? Is that useful? (QM)
- If you can't differentiate the step function, make some new mathematics where you can
- Integration out into the complex plane
- y + ay' + by'' = h(x): Motion, mass-spring, electric circuit
- Feynman sine/tan notation
- The room where I lived, Electromagnetism, Advection, Diffusion, Wave-mechanics, Quantum Mechanics
The Tower of Mathematical Modelling and Statistics
What is your best guess? How certain are you? What assumptions are necessary for your conclusions? How robust is your conclusion? The lines dividing mathematics from other subjects become blurry.
- Does smoking lead to lung cancer?
- Under what assumptions?
- Fewer assumptions is better
- More ways to achieve the same is better
- When you cannot gather more data, what then?
- Polling for elections (and communication of mathematics)
- Drug testing with fatal diseases (and presenting math to decision makers)
- Prayer and blood pressure (Page 2 bottom)
- Approximations through differential equations (roads)
- MP3: Signal analysis and compression (time-domain and frequency-domain) (modded by experience)
- IPCC (Report page 18)
- Electric power distribution (optimization) (N-1) (smart grid)
- Calculating air fares (ticket prices)
- Preprocessing of fMRI
- Regression (Netflix recommendation)
- Building bridges (Finite Element Method) (How to add wind, earthquake, soil erosion?)
- Computational Fluid Dynamics
- http://www.bbc.com/future/story/20150209-the-network-that-runs-the-world (Use pictures)
- Concave utility implies risk aversion
- Studying genes
- Improving our intuition (ex: 50 fold a paper)
- The room where I live, Machine Learning & Statistics & Data
(1) Even the famous Galileo is affected by this:
"The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word."
Comparing todays mathematical tools to those at Galileo's time is like comparing modern day building techniques using cranes to building stuff by hand. If we are amazed at what they were able to do hundreds of years ago, it is in the same way we are amazed that they were able to build the pyramids or the stonehenge. "How could they do that without proper tools?"
Note: Details are not supplied here about the topics in the three Towers. This post was written as preparation for a talk. I can perform an hour long talk (or less) on this topic on demand.
Note: To read about the context of any picture, google the image itself.
Disclaimer: If you have studied mathematics for several years at university level, you may consider parts of what I am talking about here as Computational Chemistry, Statistical Biology, Theoretical Physics., Computer Science, or other. You may also consider Mathematical Modelling or Statistics as separate from Mathematics.
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