Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.


During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.

Epistemological Constraints to Finitism. Thought of the Day 68.0


Hilbert’s substantial philosophical claims about the finitary standpoint are difficult to flesh out. For instance, Hilbert appeals to the role of Kantian intuition for our apprehension of finitary objects (they are given in the faculty of representation). Supposing one accepts this line of epistemic justification in principle, it is plausible that the simplest examples of finitary objects and propositions, and perhaps even simple cases of finitary operations such as concatenations of numerals can be given a satisfactory account.

Of crucial importance to both an understanding of finitism and of Hilbert’s proof theory is the question of what operations and what principles of proof should be allowed from the finitist standpoint. That a general answer is necessary is clear from the demands of Hilbert’s proof theory, i.e., it is not to be expected that given a formal system of mathematics (or even a single sequence of formulas) one can “see” that it is consistent (or that it cannot be a genuine derivation of an inconsistency) the way we can see, e.g., that || + ||| = ||| + ||. What is required for a consistency proof is an operation which, given a formal derivation, transforms such a derivation into one of a special form, plus proofs that the operation in fact succeeds in every case and that proofs of the special kind cannot be proofs of an inconsistency.

Hilbert said that intuitive thought “includes recursion and intuitive induction for finite existing totalities.” All of this in its application in the domain of numbers, can be formalized in a system known as primitive recursive arithmetic (PRA), which allows definitions of functions by primitive recursion and induction on quantifier-free formulas. However, Hilbert never claimed that only primitive recursive operations count as finitary. Although Hilbert and his collaborators used methods which go beyond the primitive recursive and accepted them as finitary, it is still unclear whether they (a) realized that these methods were not primitive recursive and (b) whether they would still have accepted them as finitary if they had. The conceptual issue is which operations should be considered as finitary. Since Hilbert was less than completely clear on what the finitary standpoint consists in, there is some leeway in setting up the constraints, epistemological and otherwise, an analysis of finitist operation and proof must fulfill. Hilbert characterized the objects of finitary number theory as “intuitively given,” as “surveyable in all their parts,” and said that their having basic properties must “exist intuitively” for us. This characterization of finitism as primarily to do with intuition and intuitive knowledge has been emphasized in that what can count as finitary on this understanding is not more than those arithmetical operations that can be defined from addition and multiplication using bounded recursion.

Rejecting the aspect of representability in intuition as the hallmark of the finitary; one could take finitary reasoning to be “a minimal kind of reasoning supposed by all non-trivial mathematical reasoning about numbers” and analyze finitary operations and methods of proof as those that are implicit in the very notion of number as the form of a finite sequence. This analysis of finitism is supported by Hilbert’s contention that finitary reasoning is a precondition for logical and mathematical, indeed, any scientific thinking.

Categorial Logic – Paracompleteness versus Paraconsistency. Thought of the Day 46.2


The fact that logic is content-dependent opens a new horizon concerning the relationship of logic to ontology (or objectology). Although the classical concepts of a priori and a posteriori propositions (or judgments) has lately become rather blurred, there is an undeniable fact: it is certain that the far origin of mathematics is based on empirical practical knowledge, but nobody can claim that higher mathematics is empirical.

Thanks to category theory, it is an established fact that some sort of very important logical systems: the classical and the intuitionistic (with all its axiomatically enriched subsystems), can be interpreted through topoi. And these possibility permits to consider topoi, be it in a Noneist or in a Platonist way, as universes, that is, as ontologies or as objectologies. Now, the association of a topos with its correspondent ontology (or objectology) is quite different from the association of theoretical terms with empirical concepts. Within the frame of a physical theory, if a new fact is discovered in the laboratory, it must be explained through logical deduction (with the due initial conditions and some other details). If a logical conclusion is inferred from the fundamental hypotheses, it must be corroborated through empirical observation. And if the corroboration fails, the theory must be readjusted or even rejected.

In the case of categorial logic, the situation has some similarity with the former case; but we must be careful not to be influenced by apparent coincidences. If we add, as an axiom, the tertium non datur to the formalized intuitionistic logic we obtain classical logic. That is, we can formally pass from the one to the other, just by adding or suppressing the tertium. This fact could induce us to think that, just as in physics, if a logical theory, let’s say, intuitionistic logic, cannot include a true proposition, then its axioms must be readjusted, to make it possible to include it among its theorems. But there is a radical difference: in the semantics of intuitionistic logic, and of any logic, the point of departure is not a set of hypothetical propositions that must be corroborated through experiment; it is a set of propositions that are true under some interpretation. This set can be axiomatic or it can consist in rules of inference, but the theorems of the system are not submitted to verification. The derived propositions are just true, and nothing more. The logician surely tries to find new true propositions but, when they are found (through some effective method, that can be intuitive, as it is in Gödel’s theorem) there are only three possible cases: they can be formally derivable, they can be formally underivable, they can be formally neither derivable nor underivable, that is, undecidable. But undecidability does not induce the logician to readjust or to reject the theory. Nobody tries to add axioms or to diminish them. In physics, when we are handling a theory T, and a new describable phenomenon is found that cannot be deduced from the axioms (plus initial or some other conditions), T must be readjusted or even rejected. A classical logician will never think of changing the axioms or rules of inference of classical logic because it is undecidable. And an intuitionist logician would not care at all to add the tertium to the axioms of Heyting’s system because it cannot be derived within it.

The foregoing considerations sufficiently show that in logic and mathematics there is something that, with full right, can be called “a priori“. And although, as we have said, we must acknowledge that the concepts of a priori and a posteriori are not clear-cut, in some cases, we can rightly speak of synthetical a priori knowledge. For instance, the Gödel’s proposition that affirms its own underivabilty is synthetical and a priori. But there are other propositions, for instance, mathematical induction, that can also be considered as synthetical and a priori. And a great deal of mathematical definitions, that are not abbreviations, are synthetical. For instance, the definition of a monoid action is synthetical (and, of course, a priori) because the concept of a monoid does not have among its characterizing traits the concept of an action, and vice versa.

Categorial logic is, the deepest knowledge of logic that has ever been achieved. But its scope does not encompass the whole field of logic. There are other kinds of logic that are also important and, if we intend to know, as much as possible, what logic is and how it is related to mathematics and ontology (or objectology), we must pay attention to them. From a mathematical and a philosophical point of view, the most important logical non-paracomplete systems are the paraconsistent ones. These systems are something like a dual to paracomplete logics. They are employed in inconsistent theories without producing triviality (in this sense also relevant logics are paraconsistent). In intuitionist logic there are interpretations that, with respect to some topoi, include two false contradictory propositions; whereas in paraconsistent systems we can find interpretations in which there are two contradictory true propositions.

There is, though, a difference between paracompleteness and paraconsistency. Insofar as mathematics is concerned, paracomplete systems had to be coined to cope with very deep problems. The paraconsistent ones, on the other hand, although they have been applied with success to mathematical theories, were conceived for purely philosophical and, in some cases, even for political and ideological motivations. The common point of them all was the need to construe a logical system able to cope with contradictions. That means: to have at one’s disposal a deductive method which offered the possibility of deducing consistent conclusions from inconsistent premisses. Of course, the inconsistency of the premisses had to comply with some (although very wide) conditions to avoid triviality. But these conditions made it possible to cope with paradoxes or antinomies with precision and mathematical sense.

But, philosophically, paraconsistent logic has another very important property: it is used in a spontaneous way to formalize the naive set theory, that is, the kind of theory that pre-Zermelian mathematicians had always employed. And it is, no doubt, important to try to develop mathematics within the frame of naive, spontaneous, mathematical thought, without falling into the artificiality of modern set theory. The formalization of the naive way of mathematical thinking, although every formalization is unavoidably artificial, has opened the possibility of coping with dialectical thought.

Conjuncted: Internal Logic. Thought of the Day 46.1


So, what exactly is an internal logic? The concept of topos is a generalization of the concept of set. In the categorial language of topoi, the universe of sets is just a topos. The consequence of this generalization is that the universe, or better the conglomerate, of topoi is of overwhelming amplitude. In set theory, the logic employed in the derivation of its theorems is classical. For this reason, the propositions about the different properties of sets are two-valued. There can only be true or false propositions. The traditional fundamental principles: identity, contradiction and excluded third, are absolutely valid.

But if the concept of a topos is a generalization of the concept of set, it is obvious that the logic needed to study, by means of deduction, the properties of all non-set-theoretical topoi, cannot be classic. If it were so, all topoi would coincide with the universe of sets. This fact suggests that to deductively study the properties of a topos, a non-classical logic must be used. And this logic cannot be other than the internal logic of the topos. We know, presently, that the internal logic of all topoi is intuitionistic logic as formalized by Heyting (a disciple of Brouwer). It is very interesting to compare the formal system of classical logic with the intuitionistic one. If both systems are axiomatized, the axioms of classical logic encompass the axioms of intuitionistic logic. The latter has all the axioms of the former, except one: the axiom that formally corresponds to the principle of the excluded middle. This difference can be shown in all kinds of equivalent versions of both logics. But, as Mac Lane says, “in the long run, mathematics is essentially axiomatic.” (Mac Lane). And it is remarkable that, just by suppressing an axiom of classical logic, the soundness of the theory (i.e., intuitionistic logic) can be demonstrated only through the existence of a potentially infinite set of truth-values.

We see, then, that the appellation “internal” is due to the fact that the logic by means of which we study the properties of a topos is a logic that functions within the topos, just as classical logic functions within set theory. As a matter of fact, classical logic is the internal logic of the universe of sets.

Another consequence of the fact that the general internal logic of every topos is the intuitionistic one, is that many different axioms can be added to the axioms of intuitionistic logic. This possibility enriches the internal logic of topoi. Through its application it reveals many new and quite unexpected properties of topoi. This enrichment of logic cannot be made in classical logic because, if we add one or more axioms to it, the new system becomes redundant or inconsistent. This does not happen with intuitionistic logic. So, topos theory shows that classical logic, although very powerful concerning the amount of the resulting theorems, is limited in its mathematical applications. It cannot be applied to study the properties of a mathematical system that cannot be reduced to the system of sets. Of course, if we want, we can utilize classical logic to study the properties of a topos. But, then, there are important properties of the topos that cannot be known, they are occult in the interior of the topos. Classical logic remains external to the topos.