Abstract
Numbers: where the intentional genesis of number concepts is discussed, together with the problem of “imaginary” numbers, a particular instance of the general problem of “imaginary” entities in mathematics – how things that do not exist can be useful for knowing things that exist?
Scientific subject-matter and procedures grow out of the direct problems and methods of the common sense.
John Dewey, Logic, the Theory of Inquiry
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Notes
- 1.
Husserl 1970.
- 2.
But, as Husserl himself repeatedly emphasized, psychological investigations are not without phenomenological interest, to the extent that a priori psychological inquiries into the mental life of an ego considered simply as such, which Husserl called pure phenomenological psychology, are translatable by a change of perspective into a priori phenomenological investigations of the intentional life of the transcendental ego.
- 3.
The infinite numbers introduced by Cantor, which will not concern me in this chapter, pose intriguing questions of their own.
- 4.
Although PA is anterior to his phenomenological period (from the Logical Investigations of 1900–1901 onward) Husserl never disowned it; rather, he often quoted approvingly from it, implying that it contained a correct phenomenology of arithmetic, if its more “naturalistic” aspects were given a proper phenomenological reading. I think Husserl would approve most of the conclusions I draw here, but maybe not all of them; in particular, I do not think that he would agree with my interpretation of contentual arithmetic, that is, the arithmetic of numbers proper, not things that behave only formally like numbers. For Husserl, contentual arithmetic is formal because its objects are forms; for me, albeit numbers are indeed forms, arithmetic is material in the sense that its object-matter is a particular ontological region (that of numbers) and formal in the sense that only formal properties of numbers are of concern to it.
- 5.
Recall that Husserl’s theory of intuition is far richer than Kant’s. Whereas for the latter an intuition is an individual representation, for the former just about anything can be intuited, real or ideal, abstract or concrete objects, concepts, essences, forms, you name it. Intuition means presentification to consciousness. Intuitive contents are not “representations” in the sense of “copies” of things, but things themselves – when the ego intuits something, it is this thing itself that comes before the ego’s consciousness, not a copy of it.
- 6.
Abstraction, I recall, is a sort of “refocusing” of consciousness. However, we must be careful to not read these terms on a psychological key (as is often the case). As already explained, to abstract is an operation (an act) by means of which the ego becomes aware of an aspect of an object (color, shape, quantitative form, etc.) based on its awareness of the object itself (an important form of abstraction is formal abstraction, in which form becomes salient and matter is obliterated). In abstractive acts we do not “separate” the aspect we focus on from its support (the object it belongs), either in reality, which would be absurd, or “in our heads”, that is, mentally. We simply make this aspect itself, which exists together with its support, the focal point of consciousness (and thus refer to it, theorize about it, etc.).
- 7.
To constitute or to posit are “acts” or “experiences” in which objects (or, to use a Husserlian neologism, “objectualities”), with their sense of being, come into the sphere of consciousness of the ego, that is, come into being. Constitutive acts can be iterated, the object of one serving as the matter for another. Numbers can be either intuited or merely intended, but still conceived as in principle capable of being intuited. Intuitability is equivalent to existence, but each domain of being has its own characteristic sense of intuition (for example, sensorial perception for the empirical world, mathematical intuition for the mathematical realm). Whole domains of objects can also be constituted, albeit non-intuitively, simply as extensions of concepts, without the ego having to constitute each of these objects individually, for example, the domain of real numbers. And remember, domains of being can only acquire a sense of being in intentional consciousness.
- 8.
Truth means, in this case, truth according to the concept, i.e. conceptual truth. It can be intuitive or non-intuitive in case it is merely a logical consequence of intuitive truths. The phenomenological theory of truth is still the correspondence theory. In the most basic sense of truth, an assertion is intuitively or evidently true if it is adequately filled by a correspondent intuition. Since there are various degrees of adequacy, there are various degrees of intuitiveness of truth. Notice that the correspondence alluded to in the phenomenological conception of intuitive truth is entirely confined to the intentional field. An assertion can also be true, although not intuitively, if it follows from other truths by logical derivation; truth can flow down logical chains of reasoning (and falsity up), but not intuitiveness.
- 9.
Conceptual intuition does not necessarily give us adequate access to a concept; one can have incomplete or imperfect intuitions. One cannot always expect to obtain a complete set of intuitive truths on the basis of which all conceptual questions are logically decidable, sometimes because our intuitions are inadequate, sometimes because the positing itself is incomplete.
- 10.
Positing also grounds reference. A symbol is given objective directness by the ego’s intention of using it as a name of something. Taking a symbol as a denoting symbol of an object is intending this object and, simultaneously, the symbol as a sign of it. One can use a denoting symbol without being conscious of what it means or denotes (meaning can become “fossilized”), but one can always recover both meaning and object meant by reenacting the original positing act (in which the object appears with the meanings it has). The very possibility a priori of reactivating a positing act is enough to grant the object posited objective existence. However, one can still use symbols meaningfully without knowing what they mean by abiding to objectively available rules of use originally rooted in intentional meaning. In causal theory of reference, causal chains guarantee reference; in the phenomenological theory, the possibility a priori of reactivation grants reference, even when symbols are used “blindly”, provided they are used correctly. One of the most interesting phenomena in mathematics is the subtle changes of meaning mathematical terms usually undergo as the ego continues to use the same terms to denote different, sometimes subtly different, although usually related intentional objects (for instance, number, space, etc.).
- 11.
Peter Pan is portrayed in the story as a (slightly weird) real being, and real entities must conform to determinate conditions of existence; they must, for instance, occupy a position in space and time, be able to participate in causal chains ending in the stimulation of my nervous system, and the like. Peter Pan, however, does not satisfy these conditions and thus the internal consistency of the positing act is not sustainable and the act, consequently, loses its object-positing quality: Peter Pan is not a real entity. He is an object of fantasy because the intentional act that posits it imposes conditions of existence that it does not satisfy. Numbers, on the other hand, have different conditions of existence, namely, intuitability and consistent intentional positing.
- 12.
The objectivity of mathematical truths is reinforced by the adoption of classical logic, in which the principle of bivalence (an assertion is either true or false, in which case its negation is true) is valid. By assuming that a mathematical assertion has a definite, although maybe unknown truth-value, one assumes that mathematical “facts” are determined in themselves, although not independently as realists claim. In fact, bivalence only means that we assume as a principle (in the strong sense of an unconditioned fact) that any properly phrased (i.e. meaningful) assertion has an intrinsic truth-value. This only means that, from the point of view of the a priori laws regulating the combination of the syntactic and semantic categories, nothing stands in the way of this assertion being confronted with intuitions of the appropriate type.
- 13.
We can use, I believe, Husserl’s distinction between nomological and ontological sciences to clarify the difference between Plato and Aristotle’s philosophies of arithmetic. A nomological science (such as arithmetic) is a science whose unity is given by fundamental laws of essence. Ontological sciences (such as geology, for example), on the other hand, are those whose unity depends exclusively on the unity of their domains. Concepts first or objects first. The access to the domains of nomological sciences is intermediated by the concepts that unify them, those of ontological theories, on the contrary, requires objectual intuition and inductive generalization. If no essential legality presides over a domain, our knowledge of it cannot go beyond what the intuition of the domain provides, what follows logically from it, or, at best, what can be coherently built on top of it. I think that for both Plato and Aristotle arithmetic is a nomological science whose object is the intuitable concept of number. The difference is where each believe this concept should be located. For Aristotle, it is fundamentally in the empirical world, numbers exist as (possible or actual) aspects of things whereas for Plato, the extension of the concept of number is a realm of being not of this world. For both Plato and Aristotle arithmetical truths are a priori, referring however, for the latter, to actual or possible abstract aspects of the world and, for the former, to ideal Forms.
- 14.
Conceptual truths are truths about concepts; accessing such truths requires either conceptual analyses (in which case they are analytic, in Kant’s conception of analyticity) or intuitions (in which case they are synthetic). If the synthesis necessarily involves empirical intuitions, the truths are a posteriori; in case they involve only pure intuitions, they are a priori.
- 15.
See da Silva 2010.
- 16.
It is conceivable that space is not Euclidean (in fact, modern physics endows the space-time continuum with a non-Euclidian structure), but, or so it seems, 2 + 2 can only be 4, provided the meaning of numerical terms and operations is not perversely distorted to force a different conclusion. One can call into question the a priori character of geometry, but arithmetic does not seem open to this possibility.
- 17.
Frege 1894.
- 18.
The terminology is mine, not Husserl’s.
- 19.
Although closely related, there are differences between ideation and idealization. The latter is like taking the limit of a sequence, for example, idealizing the roughly spherical form of a ball as a mathematical sphere. By realizing the possibility in principle of making, in a sequence of acts, the form of the ball approach that of a perfect sphere, the ego idealizes by closing the series, it “sees” the form of the ball as a perfect sphere. Ideation is the act in which Ideas or Forms are posited; in mathematical terms, it is the equivalent of taking the quotient by an equivalence relation; i.e. seeing as the same what is only equal under a certain aspect. From the intuitive fact that each number has a successor one can idealize by positing a limit number of the series of successors, the first transfinite number; from the equivalence of equinumerous quantitative forms one can ideate the ideal entity all these forms instantiate. Idealization and ideation are creative, object-positing acts that are essential in mathematics.
- 20.
Husserl criticizes Frege for inverting this order of priority. For Frege, as we know, equinumerosity is defined in terms of one-to-one correspondence and numbers as classes of equinumerous concepts. For Husserl, this is phenomenologically inadequate for, he thinks, the expressions “A and B have the same number” and “A and B are equinumerous” are co-extensional but not synonymous, in the sense of having the same meaning; therefore, one cannot define number in terms of equinumerosity. The former must be defined (that is, characterized as to its essence) independently of the latter, which is only a criterion of identity of number. See this discussion, which includes Husserl’s critique of Frege, in chapter VII of PA.
- 21.
There is another meaning of “formal science”, namely, a non-interpreted science concerned exclusively with abstract or ideal formal domains (or structures), not particular domains of materially determined objects. This, however, is not the meaning I give to the notion here; arithmetic, considered as a science of particular ideal forms, is eo ipso concerned with a particular domain of entities.
- 22.
Plato distinguished between ideal numbers, objects of philosophical inquiries, and mathematical numbers (collections of undifferentiated units), objects of mathematical investigation. See Klein 1968 and the definitions of unit and number in Euclid’s Elements, Book VII, Def. 1 and 2.
- 23.
See Barrow 1993, chap. 2.
- 24.
Barrow 1993, p. 39.
- 25.
Barrow 1993, p. 34.
- 26.
Husserl calls this the idealization of the “and so on”. See Husserl 1969, chap. 3 § 74.
- 27.
See Miller 1982.
- 28.
Again, by intentional constitution I do not mean the psychological process of formation of representations or the epistemological process of “grasping” something that exist “out there” independently of the subject.
- 29.
Definitions in which common aspects of a multiplicity of objects are made into ideal objects are common in mathematics, where they are known as creative definitions. For example, spatial direction as that which all parallel lines have in common.
- 30.
Since material content is no longer a concern, formal consistency is all that is required.
- 31.
See Zellini 1997. The “discovery” (in fact, invention) of imaginary numbers by the Italian algebraists of the Renaissance constitutes a classical example of this process.
- 32.
- 33.
The expression n/m = r/s is then ambiguous; if the expressions on both sides of the equation denote fractions the symbol = denotes equality, otherwise, if they denote rational numbers, = denotes identity.
- 34.
It is irrelevant whether we take N and Q as sets or as collections.
- 35.
The operations on numbers as natural numbers correspond isomorphically to operations on numbers as rational numbers.
- 36.
We must be careful here, an existential assertion, for example, even if all its constants are in N, may be true in Q but not in N. For example, “there is x such that n.x = m”, n and m natural numbers.
- 37.
See the definitions in Euclid’s Elements Book X.
- 38.
Weyl 1994.
- 39.
Set-theoretical reductionism can, of course, provide a uniform logical-conceptual context of translation, in this resides its utility, not in telling what mathematical entities “really” are.
- 40.
“Versuch den Begriff der negativen Grössen in die Weltweisheit einzuführen” (1763).
- 41.
Cardano 1993, p.9.
- 42.
- 43.
See in particular da Silva 2016a and the bibliography therein.
- 44.
See Kant 1986.
- 45.
See da Silva 2016a for details.
- 46.
Explaining the applicability of mathematics to the empirical sciences, then, boils down to explaining the formal relations between formal structures discernible in experience, or idealized from it, and purely intentional mathematical structures, and under which conditions structural properties of mathematical structures can be transferred to empirical structures.
- 47.
Structuralists who deny the existence of numbers and their relevance for arithmetic must nonetheless explain why arithmetical operations are defined the way they are. If adding numbers, for instance, has nothing to do with collecting units, why is it that we have chosen to define it as if it had?
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da Silva, J.J. (2017). Numbers. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_4
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