conceptual structural ism

8/3/2019 Conceptual Structural Ism

1/37

CONCEPTUAL STRUCTURALISM

AND THE CONTINUUM

Solomon Feferman

PHILMATH INTERSEM 2010

Universit Paris Diderot-Paris 7June 8, 2010

http://math.stanford.edu/~feferman
http://math.stanford.edu/~fefermanhttp://math.stanford.edu/~fefermanhttp://math.stanford.edu/~feferman


2/37

Is the Continuum Hypothesis (CH)a Definite Mathematical Problem?

My view: No; in factit is essentially indefinite(inherently vague).

That is, the concepts of arbitrary set and functionas used in its formulation are essentially indefinite.

This comes from my general view of the nature ofmathematics, that it is humanly based and that it

deals with more or less clear conceptions ofmathematical structures; for want of a betterword, I call that view conceptual structuralism.


3/37

The Opposite Point of View:Ontological (Platonic) Realism

Under this view, Kurt Gdel, in his article What isCantors continuum problem? (1947/1964),asserted that CH is a definite mathematicalproblem, though one that may require new axiomsof set theory in order to settle it.

Gdels program(s) for new axioms: intrinsic andextrinsic.

Currently only high hopes for the extrinsicprogram.


4/37

Mathematical Structuralism

Modern mathematics dominated by structuralistviews (abstract algebra, topology, analysis;Bourbaki, category theory, etc.)

Explicit inception often credited to Dedekind.

But mathematicians have implicitly always beenstructuralists.

Mathematics is in its most general sense thescience of relationships, in which one abstractsfrom any content of the relationships.(C. F. Gauss,Werke X/1)


5/37

Mathematical Structuralism(Contd)

Hilbert, Grundlagen der Geometrie.

In the Hilbert-Frege exchange, Frege is the oddman out.

Mathematicians do not study objects, but therelations between objects; to them it is a matter ofindifference if those objects are replaced by

others, provided that the relations do notchange. ...they are interested by formalone. (Poincar, Science and Hypothesis)


6/37

Structuralist Philosophies

of Mathematics

Paul Benacerraf, What numbers could not be (1965)

Geoffrey Hellman,Mathematics Without Numbers (1989)(modal structuralism)

Michael Resnik,Mathematics as a Science of Patterns(1997) (holistic realism)

Stewart Shapiro, Philosophy of Mathematics: Structure andOntology(1997) (ante rem structuralism)


7/37

Structuralist Philosophies of Mathematics

(Contd)

Charles Chihara,A Structural Account of Mathematics(2004) (nominalistic structuralism)

Charles Parsons,Mathematical Thought and itsObjects (2008)

Daniel Isaacson, The reality of mathematics andthe case of set theory (2008) (quasi-conceptualstructuralism)

Their positions on CH


8/37

Conceptual StructuralismThesis 1

The basic objects of mathematical thought

exist only as mental conceptions, thoughthe source of these conceptions lies ineveryday experience in manifold ways(counting, ordering, matching, combining,separating, and locating in space and time).


9/37

Thesis 2

Theoretical mathematics has its source inthe recognition that these processes areindependent of the materials or objects towhich they are applied and that they are

potentially endlessly repeatable.


10/37

Thesis 3

The basic conceptions of mathematics areof certain kinds of relatively simple ideal-

world pictures which are not of objects inisolation but of structures, i.e. coherentlyconceived groups of objects interconnectedby a few simple relations and operations.

They are communicated and understoodprior to any axiomatics or systematiclogical development.


11/37

Thesis 4

Some significant features of these

structures are elicited directly from theworld-pictures which describe them, whileother features may be less certain.Mathematics needs little to get started and,once started, a little bit goes a long way.


12/37

Thesis 5

Basic conceptions differ in their degree ofclarity. One may speak of what is true in agiven conception, but that notion of truthmay only be partial. Truth in full isapplicable only to completely clearconceptions.


13/37

Theses 6 and 7

What is clear in a given conception is timedependent, both for the individual andhistorically.

Pure (theoretical) mathematics is a body ofthought developed systematically by

successive refinement and reflectiveexpansion of basic structural conceptions.


14/37

Theses 8 and 9

The general ideas of order, succession, collection,relation, rule and operation are pre-mathematical;some implicit understanding of them is necessaryto the understanding of mathematics.

The general idea of property is pre-logical; someimplicit understanding of that and of the logicalparticles is also a prerequisite to the understandingof mathematics. The reasoning of mathematics isin principle logical, but in practice relies to aconsiderable extent on various forms of intuition.


15/37

Thesis 10

The objectivity of mathematics lies in its stabilityand coherence under repeated communication,critical scrutiny and expansion by many individualsoften working independently of each other.

Incoherent concepts, or ones which fail towithstand critical examination or lead toconflicting conclusions are eventually filtered out

from mathematics. The objectivity of mathematics is a special case of

intersubjective objectivity that is ubiquitous insocial reality.


16/37

Objectivity in Social Reality

John Searle, The Construction of Social Reality(1995)

There are portions of the real world, objective

facts in the world, that are only facts by humanagreement. In a sense there are things that existonly because we believe them to exist. ...

... things like money, property, governments, andmarriages. Yet many facts regarding these thingsare objective facts in the sense that they are not amatter of [our] preferences, evaluations, or moralattitudes. (Searle 1995, p.1)


17/37

Objectivity in Social Reality:Examples

I am a citizen of the United States.

I have voted in every U.S. presidential electionsince I became eligible by age to do that.

I have a PhD in Mathematics from the University ofCalifornia.

My wife and I own our home in Stanford,California; we do not own the land on which it sits.


18/37

More Examples

Rafael Nadal won the 2008 mens Wimbledonfinals match, and the 2009 Australian Open.

In the game of chess, it is not possible to force acheckmate with a king and two knights against alone king.

There are infinitely many prime numbers.


19/37

The Basic Conceptions of Mathematicsas Social Constructions

The objective reality that we ascribe tomathematics is simply the result ofintersubjectiveobjectivity about those conceptions and not about

a supposed independent reality in any platonisticsense.

This view does not require total realism abouttruth values. It may simply be undecided under a

given conception whether a given statement has adeterminate truth value.

Example: the presidential line of succession in theU.S. government is undetermined past a certain

point.


20/37

Conceptions of Sequential Generation

The most primitive mathematical conception isthat of the positive integer sequence represented

by the tallies: I, II, III, ...

Our primitive conception is of a structure(N+, 1, Sc,


21/37

Open-ended Schematic Truthsand Definite Properties

At a further stage of reflection we may recognizethe least number principle: if P(n) is any definiteproperty of members of N+ and there is some n

such that P(n) then there is a least such n.

The schema is open-ended. What is a definiteproperty? This requires the mathematiciansjudgment.

The property, n is the number of grains of sand ina heap is not a definite property.

What about the property, GCH does not holdat n?


22/37

Reflective Elaborationof the Structure of Positive Integers

Concatenation of tallies immediately leads us tothe operation of addition, m + n, and that leads usto m n as n added to itself m times.

The basic properties of the + and operationssuch as commutativity, associativity, distributivity,and cancellation are initially recognized onlyimplicitly.

One goes on to the relations m|n, n is a primenumber, m n (mod p), etc.

Soon have a wealth of expression and interestingproblems (primes, perfect numbers, etc., etc.)


23/37


24/37

Further Reflection

Further reflection on the structure of positiveintegers led to the structure of natural numbers(N, 0, Sc,


25/37

The Unfolding of Arithmetic

There is a general notion ofunfolding of open-ended schematic systems. (Feferman, 1996)

The unfolding of a basic schematic system for thenatural numbers is equivalent in strength topredicative mathematics (Feferman, Strahm 2000).

But beyond that, the scheme of induction ought tobe accepted for any definite property P that onewill meet in the process of doing mathematics.


26/37

Multiple Sequential Generation

Finite generation under more than one successoroperation Sca where a is an element of an indexcollection A.

We may conceive of the objects of the resultingstructure as words on the alphabet A, withSca(w) = wa in the sense of concatenation.

In the case that A = {0, 1} we also conceive of thewords on A as the finite paths in the binarybranching tree.


27/37

Conceptions of the Continuum

There is no unique concept of the continuum butrather several related ones. (Feferman 2009)

To clear the way as to whether CH is a genuinemathematical problem one should avoid thetendency to conflate these concepts, especiallythose that we use in describing physical reality.

(i)The Euclidean continuum, (ii) The Hilbertiancontinuum, (iii) The Dedekind real line,(iv) The Cauchy-Cantor real line, (v) The set 2N,(vi) the set of all subsets of N, S(N).


28/37

Conceptions of the Continuum (Contd)

Not included are physical conceptions of thecontinuum, since our only way of expressing themis through one of the conceptions via geometry orthe real numbers.

Which continuum is CH about? Their identity as tocardinality assumes impredicative set theory.

Set theory erasesthe conceptual distinction

between sets and sequences.

CH as a proposition about subsets of S(N) andpossible functions (one-one sets of ordered pairs)


29/37

The Continuum in Physical Science

The argument from indispensability for substantialportions of set theory (Quine, Putnam)

The contrary evidence from case studies for thethesis that all of current scientifically applicablemathematics can be carried out predicatively(Why a little bit goes a long way..., 1993)

In fact it can all be done in a system W (forWeyl) conservative over Peano Arithmetic(Feferman and Jger 1993/1996)

What would change this?


30/37

Conceptions of Sets

Sets are supposed to be definite totalities,determined solely by which objects are in themembership relation () to them, and

independently of how they may be defined, if at all.

A is a definite totality iff the logical operation ofquantifying over A, (xA) P(x), has a determinate

truth value for each definite property P(x) ofelements of A.

A B means (x A) (x B)

Extensionality: A B B AA = B.


31/37

The Structure of all Sets

(V, ), where V is the universe of all sets.

V itself is not a definite totality, so unboundedquantification over V is not justified on this

conception. Indeed, it is essentially indefinite.

If the operation S( . ) is conceived to lead fromsets to sets, that justifies the power set axiomPow.

At most, this conception justifies KP+Pow+AC,with classical logic only for bounded statements(Feferman, t.a.)


32/37

The Status of CH

But--I believe--the assumption of S(N), S(S(N)) asdefinite totalities is philosophically justified only onplatonistic grounds.

From the point of view of conceptualstructuralism, the conception of the totality ofarbitrary subsets of any given set is essentiallyindefinite (or inherently vague).

For, any effort to make it definite violates the ideaof what it is supposed to be.


33/37

Gdels Program and CH

Gdels argument (1947/1964) for themeaningfulness of CH and the need for newaxioms to settle it.

The intrinsic program is definitively inadequate(Koellner 2000, 2006).

The extrinsic program is being vigorously pursuedwith brilliant metamathematical work (Woodin et

al.--cf. Pettitot 2009), but what are the criteria forsuccess?

Could one prove that CH is essentially indefinite?So far only circumstantial evidence (e.g., the Lvy-

Solovay theorem) on the problem of CH.


34/37

Selected References

S. Feferman (1998), In the Light of Logic, OUP.

________ (2009), Conceptions of the continuum,Intellectica 51, 169-189.

________ (2009), Whats definite, whats not?,http://math.stanford.edu/~feferman/papers/whatsdef.pdf (Slides for H. Friedman 60th birthday

conference.)
http://math.stanford.edu/~feferman/papers/whatsdef.pdfhttp://math.stanford.edu/~feferman/papers/whatsdef.pdfhttp://math.stanford.edu/~feferman/papers/whatsdef.pdfhttp://math.stanford.edu/~feferman/papers/whatsdef.pdfhttp://math.stanford.edu/~feferman/papers/whatsdef.pdf


35/37

Selected References (contd)

S. Feferman and T. Strahm (2000), The unfolding ofnon-finitist arithmetic,APAL 104, 75-96.

___________________ (t.a.), The unfolding offinitist arithmetic, Review of Symbolic Logic.

P. Koellner (2006), On the question of absoluteundecidability, Philosophia Mathematica14(2), 153

188.

J. Petitot (2009), A transcendental view of thecontinuum: Woodins conditional platonism,Intellectica 51, 93-133.


36/37

Historical Note

An earlier version of this talk was presented at theVIIIth International Ontology Conference in San

Sebastin, Oct. 1, 2008.

A form of the ten theses of conceptualstructuralism were first presented in a talk to thePhilosophy Department of Columbia University

under the title Mathematics as objectivesubjectivity. The text was circulated then butnever published. Their first publication was inConceptions of the continuum (Intellectica 2009).


37/37

The End

conceptual structural ism

Documents