Xah Math Blog
O, math, my true love, how i have alienated thee, and you being quite difficult.
random sequence, not mathematically defined
it's interesting, that the concept of random number, or random sequence, is actually undefined in math, because it is impossible to define mathematically.
The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words “let X1,…,Xn be independent random variables…”. Yet as D. H. Lehmer stated in 1951: “A random sequence is a vague notion… in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians”.
Axiomatic probability theory deliberately avoids a definition of a random sequence. Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement “let us consider a random sequence” an abuse of language.
[2018-03-19 Random sequence]
Philosophy of Math: Logicism, Formalism, Intuitionism, and Category Theory
the 3 philosophy of math of 1900s are: ① logicism. Math is pure derivation of logic. ② Hilbert's formalism. Math is just bunch of fomulaic symbols. No meaning. ③ intuitionism. Math is mind's construction.
these 3 schools are the basis of foundations of math. Each has its problems. logic became formal (i.e. symbolic) logic, which is basically formalism. They in turn, became constructivism (a variation of intuitionism), so it can run by computer.
here's my comment to mathematician john baez:
thanks for correction.
i've always wondered what's the difference of logicism and formalism. Back in 1990s, I was introduced to them by Russell. (my top 3 fav author) Tried to read about them now and then. In 2000s, wikipedia is still not great, but in past year, i read a lot math again, and i seem to get some understanding. Is my characterization roughly correct?
i never cared about intuitionism, until now, and now i very much appreciate constructivism. As i understand, it is what mechanical manipulation can actually build.
PS few days ago i tried to read wikipedia about category theory again. Again, came away as incomprehensible abstraction. Meant to write a rant about it… but here's a gist…
so i spend half hour thinking, what does abstraction mean? After all, numbers 1 2 3 ... are abstraction to begin with. But then we have equations, such as 2*x+3 = 4, which is abstraction of description of math problems. Then, abstract algebra and 1800s's math, are the 2nd stage of abstraction. e.g. abstract algebra came from the systematic formal maniputlation of equations. And likewise stuff in algebraic geometry e.g. variety, and differential geometry geometry e.g. manifold, and so on in other branches of math.
Then, what's the next level abstraction? I'm thinking, scheme and sheaf etc (which i have no idea what they are), or, the category theory stuff.
but anyway, so i was reading Wikipedia on category theory. Category theory
in other math articles, say, holomorphic function, homotopy, hilbert space, riemann sphere, homomorphism, etc i've recently read, there's a cold definition. Most of the time, i can at least understand the definition, and go on from there. (in the above examples, i also appreciate what they mean, why they are there, etc, except hilbert space.)
But the category theory, it begin with pages of pages of meta description. And am at loss. But perhaps, to appreciate it, one must first have solid understanding of various branches graduate level math?
but then, my first exposure to it is
Conceptual Mathematics: A First Introduction to Categories by F William Lawvere, Stephen Hoel Schanuel. [Buy at amazon]
in 1990s. Which claimed to be written for laymen, and useful even outside of math. I recall, after reading 1 chapter, it's going too slow, and i was feeling, “what's the point?”. I never continued after the 1 chapter. (i don't have the book now. must gave away or something long ago.)
Math, Naturalness of a Field's Conditions
So, in algebra, you have a field. It's a set and 2 functions f and g, of the form f(a,b) and g(a,b), and X nesting properties of f and g (called commutativity, associativity, distributive, invertible, etc).
So, what happens if we have more than 2 functions, 3 parameters each? So, such study is called universal algebra. (i haven't studied, but i wonder what happens there, in general. The mix of nesting of function, i gather, would create more complex concept similar to associativity and distributive, involving 3 functions, but we don't have a name.)
But, WHAT is the fundamental nature, that real number (a field) is this specific X nesting properties? What is it, that real numbers, which we consider as naturally occurring or developed, form this “field” with such specific nesting properties?
am thinking, there must be some logical answer.
to describe my questions further, for example, real number developed because, first we have counting, 1, 2, 3, then naturally we developed 0, then we have rational, which is ratio, e.g. 1/2 as cutting a pie, then we have negative numbers (from, say, I OWE YOU). And from rational we discovered irrational, as in pythagorean. So there, we have real numbers. And, addition came from simple counting. Multiplication can be considered as a short for repeated additions.
so, am guessing, addition, and multiplication (repeated addition) necessitates the commutativity, associativity, distributive, properties?
now, having written this out, it seems obvious and is the answer to my own question.
... because, by looking at the definition of field, i've always thought, it's somehow arbitrary and complicated. I'd be interested, in a systematic approach of studying structures, e.g. a set, with n operation of m-arity, starting with n=1 and m=1. Then, we develop, all possible ways of nesting n such functions of m arity, so that associativity, commutativity (order of arg), distributivity, are just 3 of the possible properties.
but i gather that, universal algebra may began like this, but actually has become a bit something else.
similar situation is group theory
Group, is much simpler than field. But if you look at definition of group, you see that, it seems also arbitrary and complex.
But then, if you look at symmetry, such as symmetry of polyhedra, you see that all the requirement of group is necessary, and no more, no less.
but the question remain, why is the group definition seems arbitrary?
i mean, is there some point of view, so the associativity
(a • b) • c = a • (b • c)
requirement, dissolved as if it is natural?
again, i really like to see, a combinatorial exploration of all such possible condition of n functions of m arity. (as in universal algebra)
perhaps after seeing that, then one can judge, comparatively, whether the associativity condition is natural.
now, to be sure, function with 1 or 2 arity is actually perhaps the most natural. And 1 or 2 functions is also pretty bare. As opposed to, a structure with function that has 3 arity, or more than 2 functions.
If you just have 1 function, or just function wit 1 arity, than it may become too simple to have interesting things going on.
but again, would like to see a systematic combinatorial list of the conditions that may arise of n functions and m arity.
for those of you programers doing big data or AI, understand probability is essential.
major issues of writing 2D math plotter
when writing 2D math plotter, major issues are
- ① adoptive sampling. Else, you get kinks at sharp turns.
- ② divide by zero or ∞. e.g. plot 1/x.
- ③ asymptotes. e.g. in hyperbola
- ④ auto canvas range. (find max/min and/or find point cluster)
Chinese number names
- 10^52 恒河沙 (sads of eternal river)
- 10^56 阿僧祇 (asamkhya)
- 10^60 那由他 (let it go)
- 10^64 不可思議 (unfathomable)
- 10^68 無量大數 (unmeasurable big number)
Quality Free and Legal Math Textbooks https://www.patreon.com/posts/17392806
added a number theory book.
ideal of ring theory
For an arbitrary ring ( R , + , ⋅ ), let (R,+) be its additive group.
A subset I is called a two-sided ideal (or simply an ideal) of R if it is an additive subgroup of R that “absorbs multiplication by elements of R.”
- (I,+) is a subgroup of ( R , + )
- ∀ x ∈ I , ∀ r ∈ R : x ⋅ r , r ⋅ x ∈ I
digging out my 3 Volumes classic 〈Mathematical Thought From Ancient to Modern Times〉. I haven't read for 20 years. To read about how sin exp were extended to complex plane.
See also: Math and Geometry Books
Origami Simulator, by Amanda Ghassaei, at http://www.amandaghassaei.com/projects/origami_simulator/
added more logic book. Free Math Textbooks
The sound of space-filling curves: examples
this is incredibly beautiful, not just the novelty, but MUSICALLY!
[The sound of space-filling curves: examples By Herman Haverkort. At http://www.win.tue.nl/~hermanh/doku.php?id=sound_of_space-filling_curves ]
listen to the first one, the hilbert curve (the one “without intro”)
that's the best one. The other ones are not so good.
The author, Herman Haverkort, is a mathematician specializing on algorithms.
Free Math Textbooks (added books on logic)
Complex Function Grapher
local copy Complex Function plotter
[Drawing fractal Droste images By Roy Wiggins. At http://roy.red/fractal-droste-images-.html ]
Groups and Their Graphs
i learned group theory from this book, in 1997. Get it.
For coders, if you don't know math, the most important and basic of math, is group theory. (other than basic calculus) learn it to start to get into serious math.
group theory, is something you can learn the basics in 1 hour. And you can spend 5 years getting phd specialization. It is also, one of the most simple and beautiful theory, and the most important, practical, widely used.
this book i used back in 1992, and loved it.
get old edition, as newer edition of math text book don't add much. are basically scams to get you buy new.
i also like Abstract Analysis, Andrew Gleason
old article The Problems of Traditional Math Notation
hit hackernews https://news.ycombinator.com/item?id=15631151
Automatic differentiation and differentiation without limits
from ~2017-10-17 https://plus.google.com/+XahLee/posts/hzPVxNEWbe1
Spivak's Calculus on Manifolds, why is Wikipedia linking to stolen pdf?
is Spivak's Calculus on Manifolds in public domain now? why's Wikipedia linking to pdf?
3dxm, worst math software, but great for visualization of manifold
3dxm (aka 3d explore math), by my professor friends Richard Palais and Karcher Hermann, the worst software possible in the universe.
painful to use beyond comprehension
however, it is one of the best for visualizing manifold.
to see what the software can do, see
you can download 3dxm at http://3d-xplormath.org/index.html
sagemath, and the damnation of the open source
sagemath. sad. broken website. missing pictures. https://wiki.sagemath.org/art?action=show&redirect=pics
download pages looks like 1999. http://www.sagemath.org/download.html
the damnation of the open source
i heard his story of going commercial. (even retweeted it last year on twitter)
i've heard of sagemath for 10 years. I thought it's the great open source math program. Even recommend to clueless newbies. Never used it though. But only started to do math again now, so i tried to download for the first time.
and, the first impression, of its website and download page, is very sad. it reminds me all the pain of the open source.
first i go to go the website. Check out the pic demo. Sadly, 80% of pictures don't show.
i went to download page, to download anyway. I was expecting one single download button as software sites are today, which it'll just download whatever version of my os. But no, i got this wtf list of mirror sites of 1999. Quite confusing, and Virus is on my mind. And, the Mac download page has a app and non-app version, with explanation about which to download. That's one complete screwup.
i haven't tried to launch it yet. But so far, it's not encouraging. At this point, it's probably a correct guess, that if you need math lightly, just goto wolframalpha. And if you use math heavily, just $300 to Mathematica.
i wish sagemath is good, i hope it is. But open source, i think it's a sad pipe dream failure.
i hope he's successful though.
Riemannian Geometry and Mathematical Physics
Got a gift from John Baez:
GAUGE FIELDS, KNOTS AND GRAVITY by John Baez, Javier P Muniain, 1994.
excellent book. the book is really about the math of physics. More specifically, Riemannian Geometry.
the book is fast easy reading!
i think i start right at chapter 4 on differential forms.
then resolve the mysteries of stoke's theorem, exterior differential forms, cohomology, lie group, then,
in part 2 will be lots goodies for me. bundles and connection, homology. ... chern classes in chapter 4, ... and more differential geometry goodies in Part 3.
differential geometer Richard S Palais' Books and Papers http://vmm.math.uci.edu/PalaisPapers/
The Life and Mathematics of Shiing-Shen Chern
one of the greatest differential geometer of the century.
Just read this. Soul touching.
to read bio of Chern, is to also have a glimpse of modern history of China, thru the tumultuous times of war.
(watch great movie Farewell My Concubine )
and, i learned, a towering figure of differential geometry is Élie Cartan
math things i've learned.
equivalence problem. For example, triangle are defined by 3 real numbers, length of 2 of which is less than the other. So, that's the condition. It “generates” all possible triangles. And, any 2 triangle can be decided if they are equivalent (by isometry here), by first reducing the 3 numbers to certain canonical form (such as by scaling so that shortest side is 1), then simply compare the numbers.
For plane curves, it's determined by curvature function. It generates all plane curves. And to determine if 2 plane curves are equivalent by an isometry, you just express a given curve by the curvature function. Then you simply just compare 2 functions, literally.
for space curves, it's 2 functions: curvature and torsion.
for surfaces, it's the 2 fundamental forms.
the question of “equivalence problem”, is to formulate a way, so any 2 geometric object can be so compared, and unify the cases for curves and surfaces.
and Cartan began it Cartan's equivalence method
and today the method is “g structure”. G-structure on a manifold
omg, how boring is topology?
you'd think you gonna see how coffee cup turns into a donut.
instead, you got a fiat “open set”. From there on, its set of sets, subset, empty set, union of set, intersection of set, complementary set, power set, super set, finer set, coarser set, and it's set all the way!
The Method of Fluxions by Issac Newton https://archive.org/stream/methodoffluxions00newt#page/n3/mode/2up
Hilbert's 16th problem. the relative positions of the branches of real algebraic curves of degree n. Hilbert's sixteenth problem
math notation idiocy https://plus.google.com/+XahLee/posts/hzPVxNEWbe1
Comprehensive Introduction to Differential Geometry By Michael Splvak. Classic. Comprehensive Introduction Differential Geometry
that's 5 volumes heavy expensive book. and, is graduate level.
you wonder, why's such book named “introduction”. It's actually graduate text, at 5 thick heavy volumes. Mathematicians not specialized in differential geometry have lots to learn from it, i assume.
but, “introduction” implies, this 5 volumes book only touches the surface. But then, its got the word “comprehensive” in it?
so, the proper interpretation seems to be, that the topic of differential geometry really is deep, and tied with lots other advanced math. Therefore, even at 5 volumes, and being a comprehensive tome, is, still just a intro.
On several occasions, most prominently in Volume 2, Spivak “translates” the classical language that Gauss or Riemann would be familiar with to the abstract language that a modern differential geometer might use.
2017-10-08 Wikipedia Michael Spivak
that'd be interesting to read.
http://3d-xplormath.org/ for Mac
software for plotting math surfaces: Surfer
software for plotting math surfaces, especially algebraic surfaces. (Microsoft Windows, MacOS, Linux) https://imaginary.org/program/surfer
For some example of plots, see a friend Jean Constant's blog at https://jcdigitaljournal.wordpress.com/category/01-january-the-surfer-program/page/2/
Vladimir Voevodsky, Fields Medalist, Dies at 51
notation for recursion, what about notation for reduce?
been wanting a notation for recursion. Found it. In math, it's called iteration of a function, written as f^(°n). However, that doesn't cover reduce.
the term polytope is from mathematician Alicia Boole Stott. Alicia Boole Stott
understand the “expansion” operation on polyhedron
understand the “expansion” process on polyhedron. http://VirtualMathMuseum.org/Polyhedra/Icosahedron/index.html
[How real are real numbers? By Gregory Chaitin. At https://arxiv.org/pdf/math/0411418.pdf , Accessed on 2017-04-14 ] (local copy How_real_are_real_numbers_By_Gregory_Chaitin_f76de.pdf)
Learned about gyroscopic precession. Not intuitive. Don't know how the physics works out. But this video is amazing
Visualize Math with 3D Printing
Visualizing Mathematics with 3D Printing. one expensive book Buy at amazon
#physics euler's disk. haven't seen this before.
Euler's Disk Buy at amazon
this thing, your spin it, and it spins in reverse direction. Due to built up of instability, the rocking.
rattle back on amazon. Buy at amazon
Math: Density Plots of Trig Expressions. old. repost.
Kolam (Tamil-கோலம்) is a form of drawing that is drawn by using rice flour/chalk/chalk powder/white rock powder often using naturally/synthetically colored powders in Tamil Nadu, Karnataka, Andhra Pradhesh, Kerala and some parts of Goa, Maharashtra, Indonesia, Malaysia, Thailand and a few other Asian countries. A Kolam is a geometrical line drawing composed of curved loops, drawn around a grid pattern of dots. In South India, it is widely practised by female Hindu family members in front of their houses. Kolams are regionally known by different names in India, Raangolee in Maharashtra, Aripan in Mithila , Hase and Raongoli in Kannada in Karnataka, Muggulu in Andhra Pradhesh, Golam in Kerala etc., More complex Kolams are drawn and colors are often added during holiday occasions and special events.
many more, see https://twitter.com/mariawirth1/status/810098967761616897
logician Raymond Smullyan is 97 now. ☹
Raymond Smullyan is famous among armature mathematicians for his many books on logic, presented as intriguing logic puzzles.
Knights and Knaves
On a island, all inhabitants are either knights, who always tell the truth, or knaves, who always lie.
John and Bill are residents of the island.
John says: We are both knaves.
Who is what?
here's his books Raymond Smullyan books
Truly Amazing Euler, Greatest Mathematician. A Intro.
truly amazing Euler, greatest mathematician. Here's his story told by William Dunham, 2008. Save 1 hour to watch.
Best Introduction to Graph Theory
the best introduction to graph theory is this one.
if you don't know the basics, get it, read it in a week, 1 hour a day. Once you start reading, you can't put it down.
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