FOM: 87:Programs in Naturalism
Harvey Friedman
friedman at math.ohio-state.edu
Mon May 15 02:57:52 EDT 2000
This posting outlines some projects related to naturalism in the sense of
Penelope Maddy, Naturalism in Mathematics, Clarendon Press, 1997.
These projects came about partly as a result of the recent correspondence
between the four panelists on the upcoming panel discussion
Does Mathematics Need New Axioms?, ASL meeting, Urbana, June 5, 2000.
much of which has appeared on the FOM.
After writing the book
Penelope Maddy, Realism in Mathematics, Clarendon Press, 1990.
emphasizing the approach to axioms based on philosophical principles,
intrinsic justifications, and analogies with physical science, she made an
about face and wrote the 1997 book emphasizing the approach to axioms based
on an examination of goals and extrinsic justifications.
The 1997 book and the recent electronic discussions suggest certain
programmatic extensions of the naturalistic approach.
This is not meant to be any kind of blanket endorsement of the 1997 book.
In fact, the point of view here even suggests that that book is perhaps
mistitled, or is at least missing certain key disclaimers. In particular,
the content of that book as it stands is perhaps more aptly titled
"Naturalism in set theory" or "Naturalism in set theoretic mathematics." In
addition, I, personally, am still very interested in - even optimistic
about - old fashioned intrinsic justifications for set theory, including
large cardinal axioms, even though these have proved very hard to develop
in any convincing way.
1. Fragmentation.
2. Convergence.
3. Rationality.
4. Formula classes.
5. Complexity of concepts.
6. Interconnections.
7. Software.
1. FRAGMENTATION.
The idea here is that different branches of mathematics have such widely
different goals that rational practitioners may come to very different
preferences as to the choice of axioms. What may be a compelling reason for
practitioners in one branch to conclude that they need new axioms - or even
need to adopt specific new axioms - may not be a compelling reason for
practitioners of another branch.
Moreover, new reasons for needing new axioms - or adopting certain
specific new axioms - are expected to develop from time to time as a result
of intensive state of the art research. It is to be expected that the
reaction of practitioners in different branches of mathematics to such
results with regard to the issue of the need for new axioms - or the need
to adopt specific new axioms - will differ, and even remain unsettled for
some time to come.
Instead of being horrified by such fragmentation, we should welcome it. In
fact, the goal of changing the views of practitioners in various branches
of mathematics is expected to be a major catalyst for the construction of
seminal research programs of intrinsic interest resulting in a series of
ever sharper, more striking, and more relevant results.
2. CONVERGENCE.
In this vein, the most attractive goal for the mathematical logician
appears to be the demonstration that virtually the entire large cardinal
hierarchy needs to be adopted - at least in one of many essentially
equivalent forms - by all practitioners in all branches of mathematics,
pure and applied. This is what I call convergence.
It is to be hoped that convergence will be the natural outgrowth of
intensive research over the next few decades. However, it cannot be rushed,
and one will have to continually go back to the salt mines and come up with
new invigorating ideas to pull this off for all branches of mathematics -
pure and applied.
There is one very interesting aspect that may prove particularly
troublesome from some points of view. If my expectations are fullfilled,
there will be many uses of
j:V into V
for entirely natural concrete finite classification problems of the general
sort that is being carried out in Boolean relation theory, which
demonstrably cannot be replaced by, say,
j:V(lambda + 1) into V(lambda + 1),
an axiom which is believed to be entirely compatible with ZFC.
And the situation will be technically analogous to uses of
j:V(lambda + 1) into V(lambda + 1)
for entirely natural concrete finite classification problems of the general
sort that is being carried out in Boolean relation theory, which
demonstrably cannot be replaced by, say,
j:V(lambda) into V(lambda).
One attitude is that the latter results suggest that we should
i) accept j:V(lambda + 1) into V(lambda + 1) as true; or
ii) accept j:V(lambda + 1) into V(lambda + 1) as a new axiom for extrinsic
reasons.
On the other hand, the analogy between the two situations would also
suggest that we
iii) accept j:V into V as true; or
iv) accept j:V into V as a new axiom for extrinsic reasons.
But these latter two cannot be done in the framework of ZFC since j:V into
V is refutable in ZFC (Kunen, 1968).
However, one certainly doesn't want to throw out j:V into V in light of the
(anticipated) fact that it is so very useful for carrying out natural
concrete finite classification problems where lesser principles are
insufficient.
So one is naturally lead to the following device. One asserts, instead,
that there is a transitive class containing all ordinals satsifying ZF +
j:V into V. This will be equiconsistent with ZF + j:V into V.
There are two objections to this device. One is that the axiom is now
technical. The second is that there is the smell of an "instrumentalist
dodge." This is a phrase Steel uses.
Of course, it can be said that all of the j axioms are technical, in that
they involve elementary embeddings. But I have a number of results over the
years in which I give equivalent axioms (sometimes trapped between two)
which are nontechnical, using embeddings rather than elementary embeddings.
Nevertheless there is the question of coming up with a fully nontechnical
version of
there is a transitive class containing all ordinals satsifying ZF + j:V into V
that is believed to be compatible with ZFC. This can probably be done in an
entirely satisfactory way (or perhaps for something stronger, which is
fine).
In any case, we can all wait until the essential use of some form of j:V
into V is firmly established for finite classification problems of a
natural kind before we figure out exactly what the optimal form of j:V into
V is going to be.
3. RATIONALITY.
It has been suggested by set theorists such as Steel that the normal
mathematician may not be rational when it comes to the choice of new axioms
- or the choice of having no new axioms. As I have stated before on the
FOM, I do not agree with this assessment. I do, however, think that there
are signs of irrationality of a different kind on the part of normal
mathematicians.
Where I think that they are rational is in their categorical rejection of
the consideration of mathematics aimed at full generality. (It would be
entirely irrational for me as a mathematical philosopher, philosophical
mathematician, mathematical logician, foundational studies researcher,
general intellectual, University Professor, etcetera, to be uninterested in
mathematics aimed at full generality). They are concerned with arithmetic,
algebra, and geometry, which has a long tradition - far longer than
mathematical logic as it is now understood (although it is evident that
logic in the wider sense dates back to Aristotle). In fact, they are
concerned with arithmetic, algebra, and geometry at some preformal level.
This led to the rational desire to formalize arithmetic, algebraic, and
geometric reasoning for reasons that are obvious in retrospect today. In
achieving the appropriate formalizations, the normal mathematicians were
naturally led to set theory (first Cantor was led to it from series, and
later it was embraced by Dedekind, Cauchy, Weierstrass, etcetera). In
particular, to use it for the by now standard set theoretic interpretations
of arithmetic, algebra, and geometry.
This does not mean that the normal mathematician is in any way interested
in set theory per se. The normal mathematician is only interested in
arithmetic, algebra, and geometry, and to satisfy the desire/need for
rigor, the set theoretic interpretations were invented and developed.
There can be no question that these set theoretic interpretations work very
well. However, because set theory is so powerful, and contains so much that
has no (apparent) arithmetic, algebraic, or geometric meaning, it is not
all that surprising that special difficulties arise in set theory that have
(apparently) nothing to do with arithmetic, algebra, or geometry.
But the professional set theorist is likely to say: but since you use set
theory, you are tacitly assuming a world view in which there are arbitrary
sets of integers, arbitrary sets of sets of integers, etcetera. So why
aren't you concerned with their properties? After all, you believe that
they exist.
The answer is that for the normal mathematician, there is no issue of
whether they exist. There is only the issue of whether they are useful in
formalizing arithmetic, algebra, and geometry. They are not interested in
any ultimate truth about such general objects. They are only interested in
such general objects to the extent that they are useful in providing a
formalism for arithmetic, algebra, and geometry. No wonder they don't care
if logical difficulties emerge concerning the ultimate truth about such
general objects - as long as it does not affect their use in providing a
formalism for arithmetic, algebra, and geometry.
However, I question the rationality of the normal mathematician in a quite
different context. They do not seem to be interested in, or good at,
explaining in generally understandable terms to the intellectual community,
what they are about, and what they are trying to accomplish. The key
textbooks are frequently a grab bag of specialized topics which are known
to insiders to be essential tools for would be professionals. But that is
not the same as providing a coherent big picture.
4. FORMULA CLASSES.
We now come to some topics of research for the naturalist.
A working division of mathematics is provided by the 2000 Mathematics
Subject Classifcation of the AMS, at http://www.ams.org/msc/
The outer layer is copied below. Note that Logic appears as 03-XX.
00-XX General
01-XX History and biography [See also the classification number -03
in the other sections]
03-XX Mathematical logic and foundations
04-XX This section has been deleted {For set theory see 03Exx}
05-XX Combinatorics {For finite fields, see 11Txx}
06-XX Order, lattices, ordered algebraic structures [See also 18B35]
08-XX General algebraic systems
11-XX Number theory
12-XX Field theory and polynomials
13-XX Commutative rings and algebras
14-XX Algebraic geometry
15-XX Linear and multilinear algebra; matrix theory
16-XX Associative rings and algebras {For the commutative case, see 13-XX}
17-XX Nonassociative rings and algebras
18-XX Category theory; homological algebra {For commutative rings
see 13Dxx, for associative rings 16Exx, for groups 20Jxx,
for topological groups and related structures 57Txx; see also 55Nxx
and 55Uxx for algebraic topology}
19-XX $K$-theory [See also 16E20, 18F25]
20-XX Group theory and generalizations
22-XX Topological groups, Lie groups {For transformation groups, see
54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX}
26-XX Real functions [See also 54C30]
28-XX Measure and integration {For analysis on manifolds, see 58-XX}
30-XX Functions of a complex variable {For analysis on manifolds, see
58-XX}
31-XX Potential theory {For probabilistic potential theory, see 60J45}
32-XX Several complex variables and analytic spaces {For infinite-
dimensional holomorphy, see 46G20, 58B12}
33-XX Special functions (33-XX deals with the properties of functions
as functions) {For orthogonal functions, see 42Cxx; for
aspects of combinatorics, see 05Axx; for number-theoretic aspects,
see 11-XX; for representation theory, see 22Exx}
34-XX Ordinary differential equations
35-XX Partial differential equations
37-XX Dynamical systems and ergodic theory [See also 26A18, 34Cxx,
34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
39-XX Difference and functional equations
40-XX Sequences, series, summability
41-XX Approximations and expansions {For all approximation theory in the
complex domain, see 30Exx, 30E05 and 30E10; for all trigonometric
approximation and interpolation, see 42Axx, 42A10 and 42A15; for
numerical approximation, see 65Dxx}
42-XX Fourier analysis
43-XX Abstract harmonic analysis {For other analysis on topological and
Lie groups, see 22Exx}
44-XX Integral transforms, operational calculus {For fractional
derivatives and integrals, see 26A33. For Fourier transforms, see
42A38, 42B10. For integral transforms in distribution spaces, see
46F12. For numerical methods, see 65R10}
45-XX Integral equations
46-XX Functional analysis {For manifolds modeled on topological linear
spaces, see 57N20, 58Bxx}
47-XX Operator theory
49-XX Calculus of variations and optimal control; optimization
[See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX]
51-XX Geometry {For algebraic geometry, see 14-XX}
52-XX Convex and discrete geometry
53-XX Differential geometry {For differential topology, see 57Rxx. For
foundational questions of differentiable manifolds, see 58Axx}
54-XX General topology {For the topology of manifolds of all dimensions,
see 57Nxx}
55-XX Algebraic topology
57-XX Manifolds and cell complexes {For complex manifolds, see 32Qxx}
58-XX Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx,
32Wxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory,
see 49Q15}
60-XX Probability theory and stochastic processes {For additional
applications, see 11Kxx, 62-XX, 90-XX, 91-XX,92-XX, 93-XX, 94-XX]
62-XX Statistics
65-XX Numerical analysis
68-XX Computer science {For papers involving machine computations and
programs in a specific mathematical area, see Section -04 in
that area}
70-XX Mechanics of particles and systems {For relativistic mechanics,
see 83A05 and 83C10; for statistical mechanics, see 82-XX}
73-XX This section has been deleted {For mechanics of solids, see 74-XX}
74-XX Mechanics of deformable solids
76-XX Fluid mechanics {For general continuum mechanics, see 74Axx, or
other parts of 74-XX}
78-XX Optics, electromagnetic theory {For quantum optics, see 81V80}
80-XX Classical thermodynamics, heat transfer {For thermodynamics of
solids, see 74A15}
81-XX Quantum theory
82-XX Statistical mechanics, structure of matter
83-XX Relativity and gravitational theory
85-XX Astronomy and astrophysics {For celestial mechanics, see 70F15}
86-XX Geophysics [See also 76U05, 76V05]
90-XX Operations research, mathematical programming
91-XX Game theory, economics, social and behavioral sciences
92-XX Biology and other natural sciences
93-XX Systems theory; control {For optimal control, see 49-XX}
94-XX Information and communication, circuits
97-XX Mathematics education
When you click on 03-XX you get
03-00 General reference works (handbooks, dictionaries,
bibliographies, etc.)
03-01 Instructional exposition (textbooks, tutorial papers, etc.)
03-02 Research exposition (monographs, survey articles)
03-03 Historical (must also be assigned at least one classification
number from Section 01)
03-04 Explicit machine computation and programs (not the theory of
computation or programming)
03-06 Proceedings, conferences, collections, etc.
03A05 Philosophical and critical {For philosophy of mathematics, see
also 00A30}
03Bxx General logic
03Cxx Model theory
03Dxx Computability and recursion theory
03Exx Set theory
03Fxx Proof theory and constructive mathematics
03Gxx Algebraic logic
03Hxx Nonstandard models [See also 03C62]
Only the second group above can be clicked on. It turns out that 03B30 is a
listing explicitly mentioning reverse mathematics (smile).
What are the logical complexities of the theorems proved, area by area,
throughout the classification system? What are the logical complexities of
the conjectures made, area by area, throughout the classification system?
What are the logical complexities of the most celebrated results and
conjectures, area by area, throughout the classification system? What is
the statistical distribution of these, area by area, and globally over all
areas, throughout the classification system? One needs to go into the
second and third layers. E.g., 30F45, Conformal Metrics, is at the third
level.
Here are my best guesses as to the distribution among all open questions
stated in the published literature throughout mathematics of their logical
complexity. This logical complexity is computed in terms of the least
logical complexity of a demonstrably equivalent formulation that can be
easily given by a good mathematical logician.
Of course, I am excluding open questions that are stated without implicit
or explicit cardinality restrictions; e.g., separability is virtually
always a cardinality restriction. Such statements without implicit or
explicit cardinality restrictions are normally made for convenience, and
are not meant to be principal open research problems. Counterexamples have
the Sigma form, and are counted with the Pi form.
>= Pi-0-1. 99%
>= Pi-0-2. 50%
>= Pi-0-3. 33%
>= Pi-1-1. 15%
>= Pi-1-2. 5%
>= Pi-1-3. 1%
Higher. 05%.
I am probably way off. But I put these down just to get the ball rolling.
Let us call a theorem essentially Pi-n-m if and only if there is Pi-n-m
consequence which implies the theorem using only elementary means that do
not involve the ideas needed to prove the theorem. Here is my best guess as
to the distribution of theorems (with the same exclusion as above) under
this complexity measure.
>= Pi-0-1. 99%
>= Pi-0-2. 20%
>= Pi-0-3. 10%
>= Pi-1-1. 1%
>= Pi-1-2. .5%
>= Pi-1-3. .1%
Higher. 05%.
In other words, in a sense, I am guessing that mathematics is
"overwhelmingly" Pi-0-1.
Let us call a theorem or conjecture absolute if it can be seen to hold or
fail independently of what transitive model of ZFC containing all ordinals
that one is looking in, on general principles without using ideas in a
proof. What percentage of theorems and conjectures are absolute in this
sense? Again, with the same exclusion. I would guess over 99.99%.
5. COMPLEXITY OF CONCEPTS
It is well recognized that mathematical definitions are made on an
hierarchical basis, with definitions piled on top of one another. One can
trace a complicated concept like Banach algebra back down to set theoretic
primitives.
Such tracings back are called (by me) concept trees. Once one fixes on the
primitives - set theory, or some sugared form of set theory - one can
construct these trees, and they are fairly robust. To ensure robustness,
one could enlarge the primitives somewhat to include, say, very elementary
concepts such as real numbers - so one does not have to distinguish
between, say, Dedekind reals and Cauchy reals. Perhaps a good rule is,
instead, to pick among such choices according to the one that yields the
simplest concept tree.
In any case, the project is to create these concept trees associated with
areas of mathematics. Perhaps first associated with basic texts in
mathematics. One might keep track of the number of vertices, number of
distinct vertices, and above all, the depth of the trees.
Another interesting thing to keep track of is the logical formulas that are
actually used in definitions. These are rather simple, because once
definitions get too involved, one tends to create additional concepts with
additional definitions. One should study the distribution of the logical
formulas that arise in actual mathematical definitions.
Once one has fairly robust concept trees for various areas of mathematics,
one can compare the depths of those concept trees, and start commenting on
the relative depths of the areas of mathematics (smile, smile, smile).
Also, how do the shape of the concept trees (size included) change in time
as an area develops?
6. INTERCONNECTIONS.
Papers cite other papers. Papers in certain areas cite papers in other
areas. One can try to analyze the citations in terms of areas. One can then
move to define measures of the relevance of areas on other areas, and the
relevance of areas to mathematics as a whole.
7. SOFTWARE.
At some stage in the development of these projects, one could benefit
greatly from special purpose software.
**********
This is the 84th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones are:
**********
This is the 87th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones are:
1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM.
2:Axioms 11/6/97.
3:Simplicity 11/14/97 10:10AM.
4:Simplicity 11/14/97 4:25PM
5:Constructions 11/15/97 5:24PM
6:Undefinability/Nonstandard Models 11/16/97 12:04AM
7.Undefinability/Nonstandard Models 11/17/97 12:31AM
8.Schemes 11/17/97 12:30AM
9:Nonstandard Arithmetic 11/18/97 11:53AM
10:Pathology 12/8/97 12:37AM
11:F.O.M. & Math Logic 12/14/97 5:47AM
12:Finite trees/large cardinals 3/11/98 11:36AM
13:Min recursion/Provably recursive functions 3/20/98 4:45AM
14:New characterizations of the provable ordinals 4/8/98 2:09AM
14':Errata 4/8/98 9:48AM
15:Structural Independence results and provable ordinals 4/16/98
10:53PM
16:Logical Equations, etc. 4/17/98 1:25PM
16':Errata 4/28/98 10:28AM
17:Very Strong Borel statements 4/26/98 8:06PM
18:Binary Functions and Large Cardinals 4/30/98 12:03PM
19:Long Sequences 7/31/98 9:42AM
20:Proof Theoretic Degrees 8/2/98 9:37PM
21:Long Sequences/Update 10/13/98 3:18AM
22:Finite Trees/Impredicativity 10/20/98 10:13AM
23:Q-Systems and Proof Theoretic Ordinals 11/6/98 3:01AM
24:Predicatively Unfeasible Integers 11/10/98 10:44PM
25:Long Walks 11/16/98 7:05AM
26:Optimized functions/Large Cardinals 1/13/99 12:53PM
27:Finite Trees/Impredicativity:Sketches 1/13/99 12:54PM
28:Optimized Functions/Large Cardinals:more 1/27/99 4:37AM
28':Restatement 1/28/99 5:49AM
29:Large Cardinals/where are we? I 2/22/99 6:11AM
30:Large Cardinals/where are we? II 2/23/99 6:15AM
31:First Free Sets/Large Cardinals 2/27/99 1:43AM
32:Greedy Constructions/Large Cardinals 3/2/99 11:21PM
33:A Variant 3/4/99 1:52PM
34:Walks in N^k 3/7/99 1:43PM
35:Special AE Sentences 3/18/99 4:56AM
35':Restatement 3/21/99 2:20PM
36:Adjacent Ramsey Theory 3/23/99 1:00AM
37:Adjacent Ramsey Theory/more 5:45AM 3/25/99
38:Existential Properties of Numerical Functions 3/26/99 2:21PM
39:Large Cardinals/synthesis 4/7/99 11:43AM
40:Enormous Integers in Algebraic Geometry 5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees 5/25/99 5:11PM
43:More Enormous Integers/AlgGeom 5/25/99 6:00PM
44:Indiscernible Primes 5/27/99 12:53 PM
45:Result #1/Program A 7/14/99 11:07AM
46:Tamism 7/14/99 11:25AM
47:Subalgebras/Reverse Math 7/14/99 11:36AM
48:Continuous Embeddings/Reverse Mathematics 7/15/99 12:24PM
49:Ulm Theory/Reverse Mathematics 7/17/99 3:21PM
50:Enormous Integers/Number Theory 7/17/99 11:39PN
51:Enormous Integers/Plane Geometry 7/18/99 3:16PM
52:Cardinals and Cones 7/18/99 3:33PM
53:Free Sets/Reverse Math 7/19/99 2:11PM
54:Recursion Theory/Dynamics 7/22/99 9:28PM
55:Term Rewriting/Proof Theory 8/27/99 3:00PM
56:Consistency of Algebra/Geometry 8/27/99 3:01PM
57:Fixpoints/Summation/Large Cardinals 9/10/99 3:47AM
57':Restatement 9/11/99 7:06AM
58:Program A/Conjectures 9/12/99 1:03AM
59:Restricted summation:Pi-0-1 sentences 9/17/99 10:41AM
60:Program A/Results 9/17/99 1:32PM
61:Finitist proofs of conservation 9/29/99 11:52AM
62:Approximate fixed points revisited 10/11/99 1:35AM
63:Disjoint Covers/Large Cardinals 10/11/99 1:36AM
64:Finite Posets/Large Cardinals 10/11/99 1:37AM
65:Simplicity of Axioms/Conjectures 10/19/99 9:54AM
66:PA/an approach 10/21/99 8:02PM
67:Nested Min Recursion/Large Cardinals 10/25/99 8:00AM
68:Bad to Worse/Conjectures 10/28/99 10:00PM
69:Baby Real Analysis 11/1/99 6:59AM
70:Efficient Formulas and Schemes 11/1/99 1:46PM
71:Ackerman/Algebraic Geometry/1 12/10/99 1:52PM
72:New finite forms/large cardinals 12/12/99 6:11AM
73:Hilbert's program wide open? 12/20/99 8:28PM
74:Reverse arithmetic beginnings 12/22/99 8:33AM
75:Finite Reverse Mathematics 12/28/99 1:21PM
76: Finite set theories 12/28/99 1:28PM
77:Missing axiom/atonement 1/4/00 3:51PM
78:Qadratic Axioms/Literature Conjectures 1/7/00 11:51AM
79.Axioms for geometry 1/10/00 12:08PM
80.Boolean Relation Theory 3/10/00 9:41AM
81:Finite Distribution 3/13/00 1:44AM
82:Simplified Boolean Relation Theory 3/15/00 9:23AM
83: Tame Boolean Relation Theory 3/20/00 2:19AM
84: BRT/First Major Classification 3/27/00 4:04AM
85:General Framework/BRT 3/29/00 12:58AM
86:Invariant Subspace Problem/fA not= U 3/29/00 9:37AM
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