"Substance-concepts and Function-concepts"
Jeremy Heis
University of California, Irvine
25 May 2010
Turin, Italy
Cassirer says things closely allied to the views here expressed […]: "What is here expressed is just this: that
there is a system of ideal objects whose content is exhausted in their mutual relations. The 'essence' of the
numbers is completely expressed in their positions." One might wish that Cassirer were somewhat clearer
than he is in his mode of expression (and possibly thought). (Paul Benacerraf 1960, p.162)
The investigations contained in this volume were first prompted by studies in the philosophy of
mathematics. In the course of an attempt to comprehend the fundamental concepts of mathematics from the
point of view of logic, it became necessary to analyse more closely the function of the concept itself and to
trace it back to its presuppositions. Here, however, a peculiar difficulty arose: the traditional logic of the
concept, in its well-known features, proved inadequate even to characterize completely the problems to
which the theory of the principles of mathematics led. It became increasingly evident that exact science had
here reached questions for which there existed no precise correlate in the formal language of traditional
logic. The material content of mathematical knowledge pointed back to a fundamental form of the concept
not clearly characterized and recognized within logic. In particular, investigations concerning the concepts
of the series and of the limit, the special results of which, however, could not be included in the general
exposition of this book, confirmed this view and led to a renewed analysis of the principles of concept
formation itself. (Cassirer Substance and Function, Preface)
It is, it appears to me, in fact a new and fruitful point of view, which is introduced by Russell in his
treatment of formal logic. The entire "classical" logic has concerned itself with nothing but the
subsumption of contents, with the super- and sub-ordination of the spheres of two concepts. […] syllogistic
appears overall as a particularly reactionary and inhibiting moment. Logic remains bound to the point of
view of substance and thereby to the fundamental form of the judgment of predication, while the living
scientific thought more clearly aims at the concept of function as its own systematic middlepoint. One
recognizes in this connection the value and necessity of the new foundation on which Russell is seeking to
place logic. (Cassirer "Kant und die moderne Mathematik, p.7)
It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as
constitute a progression. If they are to be anything at all, they must be intrinsically something; they must
differ from other entities as points from instants, or colors from sounds. (Russell, Principles of
Mathematics, §242)
I. Why concept formation?
Is the theory of the concept, as here developed, an adequate and faithful picture of the procedure of the
concrete sciences? (Cassirer, SF, p.11)
The process of abstraction, […] if it really took place, would consist in moving backwards. It would be a
donkey-race. The man who had gone least way in it would have the advantage, in respect of fullness and
definiteness of thinking, of the man who had gone the furthest. (T.H. Green, "Lectures on Formal
Logicians," p.193)
[A]ll [forms taken by the myth of the given] have in common the idea that the awareness of certain sorts –
and by 'sorts' I have in mind, in the first instance, determinate sense repeatables – is a primordial, nonproblematic feature of immediate experience. (Sellars, "Empiricism and the Philosophy of Mind," p.157)
The abstractive theory, as Kant saw, makes the mistake of supposing that the logical space of the concept
simply transfers itself from the objects of direct perception to the intellectual order, or better, is transferred
by the mind as Jack Horner transferred the plum. (Sellars, "Phenomenalism," p.90)
Heis, "Substance-concepts and Function-concepts"
25 May 2010
If, according to the traditional logical doctrine, the concept is merely the result of "abstraction" from a
plurality of sensory data, so has it now been shown that "similar" impressions must be placed under a
determinate rule of judging, before they – as is necessary for the process of "abstraction" – can be cognized
as similar and be comprehended in a common genus. The unity of a genus presupposes the unity of an
ideal norm, and the abstractive comparison presupposes a constructive connection. In its proper
fundamental meaning, a concept is nothing other than the consciousness of this unity of synthesis.
(Cassirer Erkenntnisproblem II, p.676)
II. What does Cassirer mean by "substance" and "function"?
A Logical contrast:
S1. The logic of substance-concepts. There is no independent logic of relations. Logic is
exhausted by syllogistic (or: syllogistic + propositional logic, if in fact the latter is irreducible to
syllogistic.)
F1. The logic of function-concepts. There is an independent logic of relations.
A Metaphysical contrast:
S2. The metaphysics of substance-concepts. No object has essential properties that are
irreducibly relational. Objects are (Aristotelian) substances. (SF, p.8)
F2. The metaphysics of function-concepts.
Some (or: every) essential properties of
(mathematical) objects are irreducibly relational. Objects are positions in relational structures.
(SF, p.41)
Epistemic contrasts:
S3. Concept formation reduces content. When we form a concept by abstraction from our
knowledge of a particular, we remove the elements that are not common with other particulars and
thereby reduce the content of our knowledge of a particular.
F3. Concept formation retains or enriches content. When we form certain concepts – like
concepts expressible as mathematical functions – and apply them to particulars, we can retain (or
even enrich) all of our knowledge of the particular. (SF, p.19-21)
S4. Substance-concepts are formed subsequently to representing the particulars that fall under
them.
F4. Function-concepts can be constructed: that is, the formation of a concept is prior to
representing the particulars that fall under it. No mathematical object is "given" independently of
concepts. (SF, p.12, 56, 112 et passim)
S5. Substance-concept atomism.
concepts or knowing any facts.
It is possible to form a concept without possessing any other
F5. Function-concept anatomism. It is not possible to form a concept without possessing any
other concepts or knowing any other facts. Concept possession has epistemic preconditions
(="functions").
S6. There are "given" sensations. There is a base level of sensations whose epistemic efficacy for
a perceiver does not depend on her having any other concepts or knowing any facts.
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Heis, "Substance-concepts and Function-concepts"
25 May 2010
F6. There are no "given" sensations. There are no sensations whose epistemic efficacy for a
perceiver does not depend on her having any other concepts or knowing any facts. Nothing is
given prior to "functions." (Einstein's Theory of Relativity, p.390)
S7. Confirmation Atomism. Concepts are applied to experience independently of confirming the
judgments that contain them. Or at least, individual judgments can be empirically confirmed in
isolation from their fellows.
F7. Confirmation Holism. Concepts are related to experience only through being in a judgment;
judgments relate to experience only in systems. (Philosophy of Symbolic Forms 3, p.461)
S8. Ontological claims are justified atomistically. We are justified in claiming that objects exist
only if we can point them out in a theory-neutral way (say, by direct acquaintance). (PK, p.63)
F8. Ontological claims are justified holistically. We are justified in claiming that objects exist
when we show that they perform "an indispensable function" in our system of science. (SF, p.83)
F5-F8 taken together: There is no knowledge independent of a system of concepts and judgments.
Contrasting Theories of Objectivity:
S9. The objectivity of substance-concepts. The objectivity of concepts and judgments consists in
there being objects to which they refer.
F9. The "functional" theory of objectivity. Objectivity rests on "functions" – on there being
common rules that lead to a unified system of judgments. (PSF 1, p.79)
S10. The copy theory of knowledge. Thinking is knowledge if it faithfully mirrors the objects it
is about. The concepts of knowledge and truth are explained in terms of the concept of an object.
F10. The functional theory of knowledge. Thinking is knowledge if it performs the functions that
make objectivity possible. The concept of an object is explained in terms of the concepts of
knowledge and truth; knowledge and truth are explained in terms of objectivity. (PSF 3, p.4-5)
III. What does the argument in Ch.1 of SF lead to the conclusions in Ch.2?
For Carnap, the words 'metaphysics' and 'ontology' are pejorative terms, applicable only to the misguided
efforts of philosophers. [...] [O]ur claims to knowledge presuppose a language or conceptual scheme. We
cannot settle ontological questions by some sort of direct confrontation with the objects concerned; our
knowledge of objects is mediated. Here he is opposing the sort of view held by Russell, which relies upon
our (alleged) direct knowledge of objects, unmediated by linguistic or conceptual structures. For Russell,
this kind of (supposed) knowledge can serve as the foundation for our conceptual scheme precisely because
it is direct, and so nonconceptual. (Peter Hylton, Quine, p.234)
The attempt to present the entirety of cognition in a systematic unity ends in final Form-concepts that bring
to expression the possible kinds of relation between contents in general. In these fundamental relations are
given the final invariants to which cognition is able to advance; therefore also the “objective” standing
[Bestand] of being is grounded in them. Because objectivity is – according to the critical analysis and
meaning of this concept – itself only another designation for the validity of determinate combinatory
connections that are to be separately discovered and are to be investigated in their structure. The task of
Erkenntniskritik consists in this, to go back from the unity of the general concept of the object to the
manifold of necessary and sufficient conditions that constitute it. In this sense the thing that cognition calls
its ‘object’ is resolved into a web of relations that are themselves held together through the highest rules
and principles. (Cassirer, "Erkenntnistheorie nebst den Grenzfragen der Logik," p.13)
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