Definitions - the Frege-Hilbert controversy

Copyright 1985 by Ralph E. Kenyon, Jr.

March 26, 1985*

The Frege-Hilbert controversy centers on differing views of what 'definition' means; Frege and Hilbert differed in how they used the term. Frege seems to have been calling Hilbert's usages illegitimate. In Frege's terminology, both of these usages could be said to express different senses. Can those senses be apprehended from a reading of the correspondence, and can they be expressed in another way so as to resolve the controversy?

First, let us see what a prominent lexicographer has to say. According to Allen Walker Read,

When Lexicographers define words, they must wrestle with the problem of dealing with the great diversity of contexts. They are bound to be basically data-oriented, but they immediately get into abstraction when they begin to fashion definitions. The notion that "a meaning" is an independent entity has been fostered by popular speech; but lexicographers know better.

When theorists talk about "univocal words", they are talking nonsense. A word that has been used more than once cannot possibly be univocal. A philosopher can make the assumption (and turn it into an assertion) that he is using a particular word "univocally", but for the real world of discourse, he is stating an impossibility. I think we must take literally Leonard Bloomfield's dictum that "every utterance of a speech-form involves a minute semantic innovation." [Language (N.Y. 1943), p. 407.] . . . the citations tend to fall into what may be called "contextual clusters". I hesitate to call these clusters "meanings", because the word meaning has been subjects to such muddled controversy. I strongly support the position of the logician Willard Quine, who has said that we should "continue to turn our back on the suppositious entities called meanings." ["The Problem of Meaning in Linguistics", The Structure of Language, Fodor and Katz, eds. (Englewood, N.J., 1961), p. 22.]1

If we are to follow this lead, we can use the citations to infer what Frege and Hilbert intended by their usages involving 'definition'. Frege and Hilbert are certainly not the first philosophers to disagree about what constitutes a definition. According to the Encyclopedia of Philosophy,

The problems of definition are constantly recurring in philosophical discussion, although there is a widespread tendency to assume that they have been solved. Practically every book on logic has a section on definition in which rules are set down and exercises prescribed for applying the rules, as if the problems were all settled. Any yet, paradoxically, no problems of knowledge are less settled than those of definition, and no subject is more in need of a fresh approach. Definition plays a crucial role in every field of inquiry, yet there are few if any philosophical question about definition (what sort of thing it is, what standards it should satisfy, what kinds of knowledge, if any, it conveys) on which logicians and philosophers agree. . . .

All the views of definition that have been proposed can be subsumed under three general types of positions, with, needless to say, many different varieties within each type. These three general positions will be called "essentialist", "prescriptive", and "linguistic", types, abbreviated as "E-type", "P-type", and "L-type", respectively. . . .

According to essentialist views, definitions convey more exact and certain information than is conveyed by descriptive statements. Such information is acquired by an infallible mode of cognition variously called "intellectual vision", "intuition", "reflection", or "conceptual analysis". Prescriptive views agree with essentialism that definitions are incorrigible, but account for their infallibility by denying that they communicate information and by explaining them as symbolic conventions. Although linguistic views agree with essentialism that definitions communicate information, they also agree with prescriptivism in that they reject claims that definitions communicate information that is indubitable. The linguistic position is that definitions are empirical (and therefore corrigible) reports of linguistic behavior.2

In this entry, Abelson goes on to propose his own view of definitions, the "pragmatic-contextual" approach, which, he argues, synthesizes the three views.

A correct theory of definition would unite the partial insights of E-type, P-type, and L-type views without relying on misleading metaphors, denying the obvious informative value of definitions, or reducting definitions to historical reports of linguistic behavior. . . . The problem is to identify a special sense of "knowledge" that is appropriate to definitions . . . This special kind of knowledge may be knowledge of how to use words effectively. Use, unlike usage, is functional. As Gilbert Ryle has observed, there are misuses and ineffective uses, but there is no such thing as a misusage or ineffective usage ("Ordinary Language", in Philosophical Review, Vol. 42 1953). Usage is what people happen to do with words and is determined by habits, while use is what should be done with words and is governed by rules. To explain the right use of a word, as distinct from merely reporting its usage, a definition must give the rules that guide us in using it. In this respect definitions are rules, rather than descriptions or reports.3

A number of rules of thumb for evaluating definitions have become canonical in the literature on the subject despite the fact that they make no clear sense in terms of any of the traditional views. The following rules can be found in practically every textbook on logic. They were first suggested by Aristotle in his Topica and have survived without change by sheer weight of tradition.


(1) A definition should give the essence or nature of the thing defined, rather than its accidental properties.
(2) A definition should give the genus and differentia of the thing defined.
(3) One should not define by synonyms.
(4) A definition should be concise.
(5) One should not define by metaphors.
(6) One should not define by negative terms or by correlative terms (e. g., one should not define north as opposite of south, or parent as a person with one or more children).4

Armed with these perspectives, let us see if we can unravel the debate. In his letter to Hilbert of 1 October 1895, Frege states:

Further, the use of symbols must not be equated with a thoughtless, mechanical procedure, . . . A mere mechanical operation with formulas is dangerous (1) for the truth of the results, and . . .

The natural way in which one arrives at a symbolism seems to me to be this: in conducting an investigation in words, one feels the broad, imperspicuous and imprecise character of word language to be an obstacle, and to remedy this, one creates a sign language in which the investigation can be conducted in a more perspicuous way and with more precision. Thus the need comes first and then the satisfaction. The contrary approach, that of first creating a symbolism and then looking for an application for it, would seem to be less benificial.5

This suggests that Frege falls into the essentialist view. He would have it that symbols fulfill communication needs. In his reply, Hilbert Agrees with Frege.

In Frege's letter to Hilbert of 27 December, 1899, we can see a distinction beginning to emerge. Hilbert has adopted an axiomatic approach to geometry in which certain terms are undefined -- point, line, plane, etc. The rules for using these terms are implicit in axioms which specify relations among these terms. By moving in this direction, Hilbert has placed himself in the camp of the prescriptivists. Frege complains about this loose use.

. . . the meaning of the words 'point', 'line', 'between' are not given, but are assumed to be known in advance. At least it seems so. But it is also left unclear what you call a point. . . .6

Here the axioms are made to carry a burden that belongs to definitions. To me this seems to obliterate the dividing line between definitions and axioms in a dubious manner, and beside the old meaning of the word 'axiom', which comes out in the proposition that the axioms express fundamental facts of intuition, there emerges another meaning but one which I can no longer quite grasp. There is already widespread confusion with regard to definitions in mathematics, and some seems to act according to the rule:

If you can't prove a proposition,
Then treat it as a definition.
In view of this, it does not seem to me a good thing to add to the confusion by also using the word 'axiom' in a fluctuating sense and similar to the word 'definition'. I think it is about time that we came to an understanding about what a definition is supposed to be and do, . . .7

Frege goes on to explicitly state what he thinks a definition should do, and how he distinguishes among definitions and axioms, etc.

I should like to divide up the totality of mathematical propositions into definitions and all the remaining propositions (axioms, fundamental laws, theorems). Every definition contains a sign (an expression, a word) which had no meaning before and which is first given a meaning by the definition. . . . The other propositions (axioms, fundamental laws, theorems) must not contain a word or sign whose sense and meaning, or whose contribution to the expression of a thought, was not already completely laid down, so that there is no doubt about the sense of the proposition and the thought it expresses. The only question can be whether this thought is true and what its truth rests on. Thus axioms and theorems can never try to lay down the meaning of a sign or word that occurs in them, but it must already be laid down.8

Frege's requirement for the definition of a term is satisfied by a particular "paradigm case", that of a single statement in which one mention of an expression occurs along with a description of necessary and sufficient conditions for using that expression. For example:

We call two lines 'parallel' if and only if they are co-planar and they have no points in common.
What Frege seems not to have realized is that his stated criteria can also be satisfied by another case. A set of statements in which one or more terms appears several times seems also to "contains a sign . . . which had no meaning before and which is first given a meaning" by the combined constraints of this set of statements.

Consider this analogy. The solution to a set of simultaneous linear equations can be explicitly stated with one equation for each coordinate. Each such single equation "defines" the value of that coordinate. However, the set of equations can be stated in their unsolved form. The values of the coordinates for the solution are no less determined, but the form of each equation no longer singles out one coordinate and gives a value for it.

Frege seems to require that definitions be of the explicit form, in which the meaning of a term is stated for each term defined. In other words, the definition of a term is a single statement contain a single mention of that term along with a meaning to be associated with it, that is, specifications for the use of that term. In distinguishing axioms from definitions, he requires that all terms in axioms previously appear in definitions.

Hilbert, on the other hand, allows the implicit form in which the meanings of each of several terms in the several coordinated axioms must be simultaneously inferred (solved for). These axioms (in Hilbert's sense) comprise an implicit definition or coordinated definitions (in Hilbert's sense).

Frege continues by specifying how he uses the term 'axiom' and how axioms are to be distinguished from definitions.

I call axioms propositions that are true but are not proved because our knowledge of them flows from a source very different from the logical source, a source which might be called spatial intuition. From the truth of the axioms it follows that they do not contradict one another. There is therefore no need for further proof.9
Let us set aside the epistemological question concerning how we might know when a particular axiom (as used by Frege) is true. Since he appeals to E-type justification for axioms, namely, in the case of geometry, our "spatial intuition", the language formed by the terms he uses is 'interpreted' by reference to our spatial intuition. This spatial intuition is a particular model, and this truth is truth relative to that particular model. This makes Frege's notions of definition and axiom semantic notions which depend upon the interpretation. Statements which are true in the interpretation will not lead to a contradiction, provided the "spatial insight" is a consistent model.

In adding another criteria for what can be considered a definition, Frege states:

The definitions, too, must not contradict one another. If they do, they are faulty. The principles of definition must be such that if we follow them no contradiction can appear.10
Since his sense of the term 'definition' seems to be expressed by the paradigm case described, we might at first think that this additional requirement applies only to the formulation of definitions. I think his admonition carries more force, and should be interpreted to mean that no contradictions can appear in subsequent theorems which depend upon the definitions as well as among definitions themselves. Consequently, for Frege, a set of "correct" definitions and "true" axioms form a system which is consistent. If a contradiction appears, he would then conclude that either one of the axioms was not "true", or one of the definitions was faulty. For this kind of "true" to agree with the philosophers naive conception of TRUTH, there must be a particular interpretation which has some special claim over others. Frege asserts as much when he appeals to our "spatial intuition".

From my point of view, it seems that, although Frege is credited with first explicating the use-mention distinction in regard to terms, he did not apply this distinction at the more general level of abstraction which we could refer to as syntactic-semantic distinction. It seems natural to generalize the use-mention distinction from terms to statements. Syntactic (mention) considerations refer to the strings of symbols or tokens and how they may be arranged. Semantic (use) considerations refer to the objects in the interpretation of these strings of symbols. Hilbert seems to be making this generalization.

You say further: 'The explanations in sect. 1 are apparently of a very different kind, for there the meanings of the words "point", "line", . . . are not given, but are assumed to be known in advance'. This is apparently where the cardinal point of misunderstanding lies. I do not want to assume anything as known in advance; I regard my explanation in sect. 1 as the definition of the concepts point, line, plane -- if one adds again all the axioms of groups I to V as characteristic marks. If one is looking for other definitions of a 'point'. e.g., through paraphrase in terms of extensionless, etc., then I must indeed opposes such attempts in the most decisive way; one is looking for something one can never find because there is nothing there; and everything gets lost and becomes vague and tangled and degenerates into a game of hide-and-seek.11

Hilbert is asserting that the set of axioms in groups I thru V "simultaneously" "define" the concepts point, line, plane, etc., and that no interpretation is to be allowed. In so doing, he expresses a sense for 'definition' and for 'axiom' which differs from Frege's. He explicitly states this when he writes

My division into explanations, definitions and axioms, which together make up definitions in your sense, . . .12

I think we need to understand Hilbert that he intends for explicit definition and axioms both, taken together, to meet Frege's requirement that a definition "contains a sign . . . which had no meaning before and which is first given a meaning" by it. Clearly Frege and Hilbert express different "senses" by the terms 'definition' and 'axiom'. If we are to use Frege's sense for 'definition' then we must state that the terms 'point', 'line', 'plane', etc., are undefined terms. I think Hilbert would agree with this more modern way of describing the situation.

. . . every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. . . . Any theory can always be applied to infinitely many systems of basic elements.13

Hilbert is "divorcing" a theory from an interpretation; he is selecting the level of syntax as the criteria for evaluating consistency. This would have been more clear had the use-mention distinction been more precisely developed and available to him at the time; he simply would have said 'terms' instead of 'concepts'.

. . . every theory is only a scaffolding or schema of terms together with their necessary relations to one another, . . .

Once the step is taken which makes this distinction, then one is "freed" to see what kinds of interpretation can satisfy the structure of the theory. In discussing the relation between truth and axioms earlier, Hilbert had stated:

. . . because for as long as I have been thinking, writing, lecturing about these things, I have been saying the exact reverse: If the arbitrarily given axioms do not contradict one another, then they are true, and the things defined by the axioms exist.
Today, we say axioms satisfying this condition are consistent. We can infer that Hilbert's sense of 'truth' in this use is what we mean by 'consistent'. It's an error to think that, for Hilbert, consistency entails truth and existence; we must infer that the sense that Hilbert expressed by his use of 'true' in this citation is the same sense for which we select the term 'consistent'. Moreover, 'the things defined by the axioms exist' should be interpreted as expressing the same sense that we would express by 'every consistent theory has a model'.

The Frege and Hilbert controversy can be accounted for by recognizing that each attached different significance to the fundamental terms 'true', 'definition', and 'axiom'. For Frege, truth is a property of statements. A definition mentions a term and specifies the properties of the object to which use of the term is to refer; a definition which is not faulty is an analytic truth. An axiom is a true proposition which specifies relations among defined terms; an axiom is a synthetic truth. An intuition, insight, interpretation, model, etc., "drives" axioms as well as definitions. Use is distinct from mention, but syntax is not distinct from semantics. For Hilbert. An axiom is a well formed statement. A definition is a set of one or more axioms which specifies the relations among one or more terms. Truth is said of axioms which do not contradict. An intuition, insight, interpretation, model, etc., may be examined to see if it satisfies a set of axioms. Use is distinct from mention, and syntax is distinct from semantics.

"I don't know what you mean by 'glory'," Alice said.

Humpty Dumpty smiled contemptuously. "Of course you don't --- till I tell you. I meant 'there's an nice knock-down argument for you!'"

"But 'glory' doesn't mean 'a nice knock-down argument,'" Alice objected.

"When I use a word," Humpty Dumpty said in a rather scornful tone, "it means just what I choose it to mean --- neither more nor less."

"The question is", said Alice, "whether you CAN make words mean so many different things."

"The question is," said Humpty Dumpty, "which is to be master --- that's all."14

Alice, I mean Frege, understands, but does not agree with Hilbert's decree that theoretical structures are to be freed from the tyranny of our preconceived interpretations.

Notes and References

  1. Allen Walker Read, "The Relation of Definitions to their Contextual Basis", a paper given before the Dictionary Society of North America, June 12, 1981, Xavier University, Cincinnati, Ohio, and reprinted in ETC.: A Review of General Semantics, Vol. 39, No. 4 (Winter, 1982)
  2. Raziel Abelson, "Definition", Encyclopedia of Philosophy Vol. 2., p. 314 (1967), Macmillan, New York
  3. 3. ibid. p. 321
  4. 4. ibid. p. 322
  5. Gottlob Frege, Letter to Hilbert of 1 October 1895
  6. Gottlob Frege, Letter to Hilbert of 27 December 1899
  7. ibid.
  8. ibid.
  9. ibid.
  10. ibid.
  11. David Hilbert, letter to Frege 29.12.1899
  12. ibid.
  13. ibid.
  14. Lewis Carroll, Through the Looking Glass
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