Monday, February 8, 2010

50 Mathematical Ideas You Really Need to Know



For Anne Hathaway, and everyone who wishes to know Mathematics and Physics (Reality) ....

Tony Crilly, Mathematical Historian at Middlesex University in UK, and former Professor at the University of Michigan and the City University in Hong Kong, wrote the wonderful book of this blog article/blarticle's title.

My 14-yr-old son gifted it to me at Christmas, along with 50 Physics Ideas You Really Need To Know. Both are wonderful introductions to their respective subjects. Very concise, and a great launching point into any of the many facets of each field. Many, but not insurmountable. I love books like this that lay fields of study out in a compact way. Neither Mathematics nor Physics are difficult, in spite of what you have been mis-taught. Yes, both take work and therefore time to understand, but not nearly as much as many would have you believe.

This page is a work in progress. I will eventually link the 50 Things which I list below, as well as key points and key people. But for now, just the list, broken out in groups of 10 for the time being, for reasons, as Dirac would have said, of symmetry:

1. Zero
2. Number systems
3. Fractions
4. Squares and Square Roots
5. Pi
6. e
7. Infinity
8. Imaginary numbers
9. Primes
10. Perfect numbers

11. Fibonacci numbers
12. Golden rectangles
13. Pascal's triangle
14. Algebra
15. Euclid's algorithm
16. Logic
17. Proof
18. Sets
19. Calculus
20. Constructions

21. Triangles
22. Curves
23. Topology
24. Dimension
25. Fractals
26. Chaos
27. The parallel postulate
- Non-Euclidean Geometry
28. Discrete geometry
29. Graphs
30. The four-colour problem

31. Probability
32. Bayes theorem
33. The birthday problem
34. Distributions
35. The normal curve
36. Connecting data
37. Genetics
38. Groups
39. Matrices
40. Codes

41. Advanced counting
42. Magic squares
43. Latin squares
44. Money mathematics
45. The diet problem
46. The traveling salesperson
47. Game theory
48. Relativity
49. Fermat's last theorem
50. The Riemann hypothesis

A graphical representation of Werner Heisenberg's Indeterminacy principle, better known as "The Uncertainty Principle", being a perfect blend of Mathematics and Physics, and one that changed the world:


Now I will re-list the 50 things, this time with the short/sweet "The condensed idea" following each four-page blurb. They were written by either the author or the editor, I'm not sure which, and go like this:

1. Zero - Nothing is quite something
2. Number systems - Writing numbers down
3. Fractions - One number over another
4. Squares and Square Roots - The way to irrational numbers
5. Pi - When the pi was opened
6. e - The most natural of numbers
7. Infinity - A shower of infinities
8. Imaginary numbers - Unreal numbers with real uses
9. Primes - The atoms of mathematics
10. Perfect numbers - The mystique of numbers

11. Fibonacci numbers - The Da Vinci Code unscrambled
12. Golden rectangles - Divine proportions
13. Pascal's triangle - The number fountain
14. Algebra - Solving for the unknown
15. Euclid's algorithm - A route to the greatest
16. Logic - the clear line of reason
17. Proof - Signed and sealed
18. Sets - Many treated as one
19. Calculus - Going to the limit
20. Constructions - Take a straight edge and a pair of compasses ...

21. Triangles -  Three sides of a story
22. Curves - Going round the bend
23. Topology - From donuts to coffee cups
24. Dimension - Beyond the third dimension
25. Fractals - Shapes with fractional dimension
26. Chaos - The wildness of regularity
27. The parallel postulate - What if parallel lines do meet?
- Euclids fifth postulate and  Non-Euclidean Geometry
28. Discrete geometry - Individual points of interest
29. Graphs -Across the bridges and into the trees
30. The four-colour problem - Four colours will be enough

31. Probability - The gambler's secret system
32. Bayes theorem - Updating beliefs using evidence
33. The birthday problem - Calculating coincidences
34. Distributions - Predicting how many
35. The normal curve - The ubiquitous bell-shaped curve
36. Connecting data - The interaction of data
37. Genetics - Uncertainty in the gene pool
38. Groups - Measuring symmetry
39. Matrices - Combining blocks of numbers
40. Codes - Keeping messages secret

41. Advanced counting - How many combinations?
42. Magic squares - Mathematical wizardry
43. Latin squares - Sudoku revealed
44. Money mathematics - Compound interest works best
45. The diet problem - Keeping healthy at any cost
46. The traveling salesperson - Finding the best route
47. Game theory - Win-win mathematics
48. Relativity - The speed of light is absolute
49. Fermat's last theorem - Proving a marginal point
50. The Riemann hypothesis - The ultimate challenge

14 comments:

Jérôme CHAUVET said...

I like that little animated picture. It could not be clearer.

Phil Warnell said...

Hi Steven,

This is indeed a nice little list that you have compiled here. You express that mathematics being not that difficult, yet requiring a little work to grasp as to understand and I would agree, yet this does not explain why so many would rather have their teeth pulled then learn something about it. I would say that like science and physics more generally those that like mathematics must first come to know the wonder of it as to how something so abstract and removed from everyday existence has none the less such a close connection with our world. The person I found to have best expressed this in my opinion is John D. Barrow, in a book he wrote in 1992 entitled ‘Pi in the Sky, Counting, Thinking, and Beingb’. It explores mathematics from the very dawn of its beginnings all the way up to current times. It examines both its foundations and advances but more importantly its power and mystery. However Barrow himself best tells you what you are in store fpr in his opening paragraph when with him saying the following:

”A mystery lurks beneath the magic carpet of science, something scientists have not been telling, something too shocking to mention except in rather esoterically refined circles: that at the root of the success of twentieth-century science there lies a deeply ‘religious belief’---a belief in an unseen and perfect transcendental world that controls us in an unexplained way, yet upon which we seem to exert no influence whatsoever. What this world is, where it is, and what it is to us is what this book is about.”

It is the understanding of what Barrow speaks of here that has those decide whether mathematics is wonderful or not as the first thing being reguire is that sense of wonder.

Best,

Phil

Steven Colyer said...

Hi Jérôme,

Yes, that is amazing. That one diagram, which I got here from page 1 Andrew Thomas' "What is Reality?" website, literally saved me from quitting MathPhys. There are several walls to overcome in Physics, and Indeterminacy is usually the first. Most don't pass, and the reason I believe so is because half the professors who teach it (per Smolin, 3RQG), don't believe it. But it's real: It is and always will be impossible to measure both the position and momentum of a fundamental particle precisely. Always, there is a price to be paid for the exactness of one in the form of a doubt of the other. Quantum mechanics is truly weird, but ... Mathematics explains it!

Hi Phil,

Thanks for that stuff about Barrow. Have you ever checked out "Philosophy of Mathematics" in the macropedia section of Encyclopedia Brittanica? Great stuff. To answer Barrow's question, I strongly feel Mathematics is discovered, not invented. The Mandelbrot Set was always there. Mathematics is NOT a science, it is a language. To call it "Science" is to diminish it, I feel. But like other languages, it is a work in progress. A wonderful journey, best summed up by Russell Crowe as John Nash IMO near the end of "A Beautiful Mind."

Is Computer Science, which SEEMS Mathematical, a Science? I say yes, for that is an Algorithmic field of study, and Algorithms are not just Logic based, but must be tied into something REAL to serve any purpose. And Reality is Science. Pure Mathematics (wonderful) is NOT real, therefore not Science.

And Logic is THE point of Philosophy. So it all gets back to Philosophy. From Logic sprang Mathematics, with Physics following. With Computer Science, Logic returns.

"Logic Development" always struck me as a better name for Philosophy, or "Logic Erudition" for the erudite elite who like big words. :-)

Computer Science and Algorithms would then be "Applied Logic". But
what do I know? I still think "Physics" should be called "Reality." Bah, humbug.

Jérôme CHAUVET said...

half the professors who teach it (per Smolin, 3RQG), don't believe it

To not believe in Heisenberg's Uncertainty Principle is misunderstanding Heisenberg's original view. "To believe" was exactly what Heisenberg did not want to regarding matter, which led him to the uncertainty principle. His paradigm was "try to be as close to reality as can be". On the contrary, Schrödinger was a dreamer, who dreamed the electron with equations.

In the end, both approaches were essential, so we have today a theory of quantum which is as complete as possible.

Best,

Steven Colyer said...

Quantum theory is not quite complete. Much work remains to be done in quantum electrodynamics and quantum chromodynamics, and then there is the dream of quantum gravity.

I liked your thoughts on Heisenberg and Schrodinger, the two bigs of QM. Max Born too, because confronted with a certain problem, he just squared to eliminate negatives, and the gift that was that phi squared equaled the electron's map of probability.

Intel is the result of that, among may others.

Phil Warnell said...

Hi Jérôme,

It depends what one means to believe in the uncertainty principle, as whether it refers to a unavoidable state of complete ignorance or an unattainable complete knowledge of actual state. To bo believe in the former forces one to also believe in either collapse scenarios or many worlds and also the measurement problem. However if one subscribes to the latter incorporating an interpretation having a dual ontology none of the other beliefs are required. The way I’ve always looked at what comprises to be good science is when it has us to know more and required to believe less and that’s why for me when it comes to the explanations of Quantum mechanics the deBroglie-Bohm one is the only one to satisfy science’s demands. Of course I’m not the first to see things this way as I’m simply relating J.S, Bell’s arguement which still goes largely ignored. In fact I would say that anyone that has not read and studied carefully the entirety of J.S. Bell’s ‘Speakable and Unspeakable in Quantum Mechanics’ has little chance to come to know why they don’t understand quantum mechanics.

Best,

Phil

Phil Warnell said...

Hi Steven,

Barrow contrasts the formalists with constructivist-empiricists and today's ultimate hackers to conclude that there remains a residue of Platonic religious mysticism in our feelings about mathematics. ""All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics 'is'...why it works nor where it works; if it fails or how it fails."" Heady stuff this, caviar for the connoisseur--but not for the innumerate.

Well if you believe that mathematics is something one discovers rather then invents then you would not be in disagreement with either Barrow or me. I would also recommend if you can find the time to read this book as I think you would enjoy it completely. As the last line taken from the that review I highlighted in my last post reminds its” heady stuff” only appreciated by the connoisseurs of the subject and so I invite you to feast.

Best,

Phil

Jérôme CHAUVET said...

Hi Phil,

I agree with you. All consequences of the uncertainty principle lead to question a lot about it, even in a weird way (e.g., the multiverse explanation). However, most of those consequences were not imagined by Heisenberg himself. I cannot imagine Heisenberg would have advanced that a multiverse could be. The latter rather comes from Schrödinger's cat paradox school, i.e., the school of those who want to see what quantum systems look like INSIDE.

In his original paper introduction, which I read, Heisenberg writes that one should focus only on what is observable,i.e., what can be measured by an EXTERNAL observer.

To me, most of those who are reasoning about the consequences of the uncertainty principle are not following Heisenberg's original line.

Hence my equation:
To not believe in the uncertainty prinple = to not believe that measuring quantum systems yields non-fully-predictable signals.

which, as we know it from experience, is wrong.

Roughly speaking, my own view about the profound nature of quantum systems is that they may have a deterministic inner density (that does not necessarily means having a probabilistic behaviour), but which however should be measured according to a probabilistic model for an external observer. A good analogy to this view is the way we pick up prime numbers (that have densities) among the naturals with probabilistic algorithm.

Best,

Jérôme CHAUVET said...

Quantum theory is not quite complete.

Heh Steve, you are out of the way! You are controversial!

Quantum theory is no more a work in progress, so we can think it is thought to be complete. Wake up!Today's paradigm is... THE SUPERSTRING THEORY (and the orchestra starts up with horns).

hope you get the irony of the tone ;)

Steven Colyer said...

Superstring Theory is unique mathematics. Pure mathematics, and not particularly beautiful rigorous math. By calling it "unique," I am insulting it. If I thought more of it (and I don't) I would call it "interesting." More yet, "intriguing." It passes none of those tests with me. I wouldn't even call it "Science Fiction." It is "Science Fantasy."

The basic flaw is the assumption: 1-dimensional strings. One needn't delve into branes, supersymmetry, tautological anthropic garbage, etc. to reject it. Who says 1-D? String theorists. Maybe it's 0.95-D. Maybe 1.1-D. Maybe all sorts of things.

When the assumption is wrong, the theory is suspect. Strings' "M-Theory" isn't a theory so much as an idea of a theory. Superstrings stand on a house of sand.

Time spent on condensate physics (or just about anything else) is time much better spent than time wasted on tautological strings.

Jérôme CHAUVET said...

Steven: That is definitely my view-point either.

I had my universitary education in experimental science, and as such, it is impossibe for me to agree with those making toy theories. That's it, superstring theory is well a toy theory, with which stringists play as if they were grown up children. It is not in any case a set of formulas aimed at talking about the nature.

In my scientific life, I have made one terrible mistake. I have read Brian Greene's "The elegant universe". Don't blame me for that. The book was fun reading it, but sorry, I had learned deeper concepts about the nature by reading Pierre Boulle's "Planet of the Apes" which unlike the former addresses a REAL philosophical issue.

I could not stop laughing after reading that it could/may/would/might be perhaps possible to maybe eventually see... some macroscopic superstrings floating around in the cosmos (it seems like one of the five interpretation of superstrings allows such a jelly stuff to occur in space-time).

I think there is nothing to add here.

With that book (a best-seller) Brian Greene has people believing that science is a fiction, in which you can invent what you please, when things do not fit with reality.

I am interested in Alain Connes' work, and I read some of his works. I read Heisenberg's work, Schrödinger's work, De Broglie work's, as I think it was important to read them. But I'll never read one single formula of this masquerade, or I had better say, of this scientifically recognized crackpottery.

Best,

Jérôme CHAUVET said...

Time spent on condensate physics (or just about anything else) is time much better spent than time wasted on tautological strings.

As long as it is not my time which is concerned, I do not care. The problem is that there is also much money and many professor positions spent on that.

Best,

Steven Colyer said...

People are leaving superstrings in droves (here in the US for sure), and in the most diplomatic way possible. Sometimes, they just drop a bombshell, which makes it fairly obvious they've left. Like Verlinde.

Please don't call superstrings "fiction", because that insults fiction. Call it fantasy. More appropriate.

(Having said that, I'm as much as for pure mathematics as the next guy. I hope superstrings continue to be studied. It sure needs more rigor. Witten can't do everything. Just ... please don't call it Physics. It's not real.)

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