HOW BIG ARE THE PARTS OF MATHEMATICS?
This collection is organized by topic along the lines of the AMS
subject classification schemes. I thought it might be interesting to
see how comprehensive the collection is by comparing to some other
objective measures of the size and scope of mathematics.
One possibility to is to poll the members of the world's mathematics
societies about what they work on. Of course this is impractical. I
believe the total membership in these organizations now reaches into 6
figures. More realistic is to poll the new PhDs in mathematics. The
US produces the lion's share of these, now running to about 1200 per
year. The AMS produces yearly (November) summaries of new PhD
production, and breaks them down into a few broad categories (in
several ways; the pigeonholing of interest here is the general topic
of the PhD). Here are the data for a couple of recent years:
95/96 92/3 82/3 (with title thru '88)
vv Logic - 25 2% 16
95 Discrete Math/Combinat. - 59 5% n/a (or part of "Computer Science")
168 Algebra & Number Theory - 158 13% 92
42 Real & Complex Analysis - 140 11% 137 "Analysis & Funct. Analysis"
171 DE,Func An, Harm Analys - ---
40 Linear&Nonlinear Optim - 54 4% 63 "Operations Research"
151 Geometry and Topology - 145 11% 92
266 Statistics - 287 22% 188
^^^ Probability - 47 4% 22
81 Numerical Analysis - 49 4% 18 "Computer Science"
121 Applied Math - 188 15% 103
18 Other - 50 4% 61
1153 TOTAL--------------------1202--- 792 (Includes Canada thru '88)
I'd like to collect cumulative data for the last decade or two. This
would give an idea of what people out in the field started with (and
usually stayed with).
An alternative perspective on "how big are the parts of mathematics"
comes from examining the recent literature. I ran some quick searches
on the Math Reviews database; Keith Dennis was most helpful in
providing raw data. The data cover essentially all published
mathematics, in my case 1980-1997; each entry has a unique Primary
Classification code xxyzz (xx, zz numeric and y a letter). I asked for
a simple count of how many reviewed articles had each of the possible
2-digit codes. (Sixty-two of those values have been in use during this
period. I think there are around 380 3-digit codes and 4900 5-digit
codes in current use. An "average" 5-digit code sees about 10 papers per
year, but this is highly variable.)
Please keep in mind that this measures only _currently produced_ research,
not fundamental or old material, and not necessarily important or quality
material. (Some comparatively accessible topics, such as elementary
number theory or calculus, tend to account for a fair number of admittedly
forgettable inclusions.)
The total count of entries so classified is about 0.8 million items.
(By comparison, there were only some 40 000 articles in mathematics
published during the entire 19th century; now, more than that are
published each year.)
Collecting the AMS codes into groups I can give partial sums. Code-by-code
totals are below (if I haven't made a transcription error).
General (codes 00-01) 23.5K articles (2.8% of the total)
Logic/Set Theory (03-04) 22.7K ( 2.7%)
Discrete math (05-08) 33.5K ( 4.0%)
Algebra/Numthy (10-22) 95.2K (11.4%)
Analysis/FuncAn (26-49,58) 220.3K (33.0%)
Geometry/Topol (51-57) 56.4K ( 6.8%)
Probability/Stat (60-62) 88.7K (10.4%)
Numerical/Comput (65-68) 74.1K ( 8.9%)
Applications (70-94) 214.2K (32.1%)
...where there seems to be some roundoff error (I'm doing this as I
write, sorry). It's interesting to compare to the PhD
percentages. There is significantly less reviewed statistics
literature than the current PhD production would suggest (as well as a
little less in topology), and significantly more literature produced
in applications, analysis, functional analysis, and optimization (as
well as a small but significantly greater production of literature in
numerical analysis and computer science.)
Here are the totals of articles in the dataset, sorted by
Primary Classification. I labelled each with asterisks, one for every
2000 items in the database.
*** 6407 00 General
********* 17136 01 History and Biography
*********** 21464 03 Mathematical logic and Foundations
* 1264 04 Set Theory
************* 26039 05 Combinatorics
*** 5436 06 Ordered structures and lattices
* 2130 08 General algebraic systems.
** 4378 10 Number Theory ---\
********* 18215 11 Number Theory ---/
** 3346 12 Field theory and polynomials
** 4968 13 Commutative rings and algebras
***** 10369 14 Algebraic Geometry
*** 6567 15 Linear and multilinear algebra, matrix theory
***** 10785 16 Associative rings and algebras
*** 6405 17 Nonassociative rings and algebras
* 2908 18 Category theory, homological algebra
* 439 19 K-Theory
********** 20366 20 Group theory and generalizations
*** 6181 22 Topological groups, Lie groups
*** 5459 26 Real functions
*** 5147 28 Measure and integration
******* 14686 39 Complex variables
* 2516 31 Potential theory
***** 10050 32 Several complex variables, analytic spaces
** 4547 33 Special functions
************* 25199 34 Ordinary differential equations
******************** 39749 35 Partial differential equations
* 2878 39 Finite differences, functional equations
* 1962 40 Sequences, series, summability
***** 9411 41 Approximations, expansions
**** 7720 42 Fourier analysis
* 2242 43 Abstract harmonic analysis
* 1834 44 Integral transforms, operational calculus
** 4246 45 Integral equations
*********** 21252 46 Functional analysis
********* 18635 47 Operator theory
****** 11949 49 Calculus of variations, optimization
*** 6291 51 Geometry
** 4789 52 Convex and discrete geometry
********* 17571 53 Differential geometry
******* 14379 54 General topology
** 4839 55 Algebraic topology
**** 8657 57 Manifolds, cell complexes
**************** 31271 58 Global analysis, analysis on manifolds
********************* 41513 60 Probability, stochastic processes
******* 13492 61
***************** 33687 62 Statistics
***************** 33867 65 Numerical analysis
******************** 40202 68 Computer science
*** 6992 70 Mechanics of particles and systems
******** 15839 73 Mechanics of solids
********* 17336 76 Fluid mechanics
** 4242 78 Optics, electromagnetic theory
* 2934 80 Thermodynamics, heat transfer
********************* 41087 81 Quantum theory
******** 16821 82 Statistical mechanics, structure of matter
* 16686 83 Relativity, gravitation
* 1240 85 Astronomy, astrophysics
* 2007 86 Geophysics
******************* 39147 90 Economics, mathematical games, O.R., L.P.
****** 12200 92 Biology, natural and behavioural sciences
************** 27516 93 Systems theory, control theory
***** 10427 94 Information, communication, circuits