# What is the philosophical ground for distinguishing logic and mathematics?

I was wondering why the field of mathematics and that of logic are perceived as two distinct fields. Although could be pleased with the intuition that logic is rather meta-mathematics, still would like to know: were there any philosophers who may be said to have created this distinction? What is the philosophical background for distinguishing mathematics and logic?

• Historical: For Kant (Preface Critique of Pure Reason B edition) logic is totally abstract as a science and has no concrete objects at all, while maths does have objects, but a priori ones. But that's 18th century and not really sufficient for the contemporary sciences. May 9, 2016 at 21:24
• @PhilipKlöcking, thank you for this helpful comment. May 9, 2016 at 21:37
• There is such a thing as mathematical logic, which is distinct from logic proper, and even in mathematics it's seems to occupy a place removed from the main current of number & geometry. Aug 13, 2016 at 20:56
• One place though where it is achieving something of a synthesis and parity is in topos theory; for example, forcing which is a technique in mathematical logic and set theory, can be given a geometric meaning using sheaf-theoretic means - but as one can see from this, we're far from logic, and deep into mathematics; which might make it perhaps problematic for philosophy. Aug 13, 2016 at 22:24
• @L.M.Student This is not a pipe. Mathematics is logical. Mathematical logic is mathematics, so it is logical. Logic is a mental capacity. We can reason logically or not, irrespective of any mathematics. So perhaps we should first distinguish logic and formal logic. Formal logic is in essence mathematics but it is not logic. It is at best a model of logic, a formal model. So, this is not a pipe. I can also faults mathematical logic for not being a correct model of logic, which also proves that mathematical logic is not logic. Apr 8, 2021 at 11:01

Let me give you a historical background first. Until the end of 19th century logic was almost exclusively associated with Aristotelian logic, the syllogistic. This logic did not have quantifiers, or even propositional variables, in other words it was too weak to support even arithmetic, let alone the rest of mathematics (Chrysippus, an ancient Stoic, and Leibniz conceived of modern propositional logic before Frege, but their ideas were disregarded and largely forgotten). It is due to this weakness that philosophers like Locke, Hume, and Kant, considered analytic knowledge, achievable through "pure logic", as entirely trivial and incapable of producing anything of substance, see Was Locke right that analytic knowledge is vacuous? Mathematics, on the other hand, clearly displayed non-trivial truths, as Euclid's work amply demonstrated, and so could not possibly reduce to logic. Kant even invented a new notion of "synthetic a priori" that rely on additional faculty of productive imagination to take mathematics beyond mere logic. Friedman in Kant's Theory of Geometry explains in detail how the lack of quantification in the syllogistic forced early calculus and analysis to rely on intuitive ideas about motion, preventing more formal constructions.

So before introduction of quantificational logic by Frege and Peirce at the end of 19th century, see Bonevac's History of Quantification, mathematics and logic did not need to be specifically distinguished, they were worlds apart. Peirce, who took early philosophical notice of the algebraization and formalization of mathematics in 19th century, believed that it does not require formal logical foundations, and that quite the opposite, logic (whose scope he understood expansively in the Kantian-Hegelian sense) depends on mathematics philosophically. It was Frege who thought the other way, and developed the technical means for reducing arithmetic (and the rest of mathematics) to logic in his ground-breaking Begriffsschrift, eine der Arithmetischen Nachgebildete Formelsprache des Reinen Denkens (Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic, 1879), and in Grundgesetze der Arithmetik (1893) outlined a programme of reducing arithmetic to logic, known as logicism. This programme indeed envisioned logic, the new mathematical logic, in a single unit with mathematics and metamathematics, both as their tool and foundation.

However, logicism quickly ran into troubles, first with the Russell's paradox, which showed some of Frege's "basic laws of thought" to be problematic (Basic Law V, the law of extensions, combined with the substitution principle implied that every predicate defines a class, which produced Russell's paradoxical class). And then when Russell attempted to remedy this in his Principia it turned out that even Frege's logic could not support all of mathematics without extra assumptions of distinctly non-logical flavor, like the notorious axiom of reducibility. The final blow to classical logicism, in its last incarnation developed by Carnap, was delivered by Gödel's incompleteness theorem, which ended the idea that an all-encompassing logical system can serve as a foundation for both mathematics and itself, see Friedman's Logical Truth and Analyticity in Carnap's "Logical Syntax of Language" for detailed discussion of subtleties involved. It ended even a more generous proposal for basing meta-mathematics on "geometry of symbols" in addition to logic, Hilbert's formalism, see Was there a Kantian influence on Hilbert's formalist programme? More recently however, some neologicist proposals were advanced by Heck and Hale-Wright.

• I'm not groking: Gödel's incompleteness might have refuted the idea of completeness of logic for describing mathematics, but why does that imply that they are separate? If anything, his insights about self-reference only serve to confirm that logic and math are indistinguishable. May 9, 2016 at 22:38
• @Alexander Logicism is about more than keeping them together, even Quine's "everything is synthetic and empirical" does that, they wanted reduction to logic. Still, It takes Friedman five pages (89-93) to explain how incompleteness undermines that:"Here is where Godel's Theorem strikes a fatal blow... Analytic-in-L fails to be captured in what Carnap calls the "combinatorial analysis"... Hence the very notion that supports, and is indeed essential to, Carnap's logicism simply does not occur in pure syntax as he understands it". May 9, 2016 at 23:39
• "If this notion is to have any place at all, then, it can only be within the explicitly empirical and psychological discipline of applied syntax; and the dialectic leading to Quine's challenge is now irresistible. In this sense, Godel's results knock away the last slender reed on which Carnap's logicism (and antipsychologism) rests". Without the analytic/synthetic distinction there is no place left for analytic "laws of pure thought", let alone reducing mathematics to them. May 9, 2016 at 23:45
• @L.M.Student I added a bit on Peirce and a comprehensive historical reference that may interest you. Boole produced the first modern algebraization of propositional logic, and de Morgan discovered the logic of relations, which is irreducible to syllogistic and heavily influenced Peirce in his introduction of modern quantification. They both thought of what they were doing as algebra, by analogy to complex numbers and quaternions, just applied to logic rather than geometry or physics. Aug 13, 2016 at 20:17
• Thank you very much for this! I am indeed interested in history of logic. The reason why asked of Boole and De Morgan is in part due to trying to pick out the differences between the trend of logic as algebra and the trend of logic as language, the latter is reflected in Frege-Russell tradition. Will look into the historical account you attached. Thank you again. Aug 13, 2016 at 20:24

The distinction between mathematics and logic was almost universally held before modern times.

Aristotle's logical works (Prior and Posterior Analytics) are part of his works that later philosophers grouped as the Organon (tool). Hence, logic was seen as a tool.

Ancients like Boethius and medievals like St. Thomas Aquinas, and logicians like John Poinsot, et al. all considered logic to be an art (the art of reasoning). E.g., Aristotle writes in Metaphysics I (980b26) that “the human race lives by art and reasonings.” St. Thomas Aquinas writes in the proem of his Expositio libri Posteriorum Analyticorum:

…an art is needed to direct the act of reasoning, so that by it a man when performing the act of reasoning might proceed in an orderly and easy manner and without error.

And this art is logic (logica), i.e., the science of reason (rationalis scientia).

Hence why logic should be taught first (Sententia Ethic., lib. 6 l. 7 n. 17 [1211.]):

[T]he proper order of learning is that boys first be instructed in things pertaining to logic because logic teaches the method of the whole of philosophy. Next, they should be instructed in mathematics, which does not need experience and does not exceed the imagination. Third, in natural sciences, which, even though not exceeding sense and imagination, nevertheless require experience. Fourth, in the moral sciences, which require experience and a soul free from passions [...]. Fifth, in the sapiential and divine sciences, which exceed imagination and require a sharp mind.

This ordering is based on the three degrees of abstraction, of which mathematics was the second. Boethius, following Aristotle, proposed that the "Speculative sciences may be divided into three kinds: physics, mathematics, and metaphysics:"

1. Physics [i.e., the natural sciences] deals with that which is in motion and material.
[ens mobile or "mobile/changeable being"]
2. Mathematics deals with that which is material and not in motion.
[∵ mathematical objects, "mathematicals," do not move or change]
3. Metascience* deals with that which is not in motion nor material.
*i.e., "metaphysics" in the Aristotelian sense of the study of "being qua being"

They are two separate fields.

What often passes as “logic” today is actually mathematical logic. Mathematical logic isn’t really logic at all, not only because its aims are different, but because it doesn’t distinguish between properly intentional signs and real signs. Mathematical logic gladly absorbs any formal structure or relation. Logic, on the other hand, is concerned with intentional signs, where the logical relation of identity is key.

What does that mean? Consider the relation between subject and predicate. This is not a real relation, as in the real, there is no relation between Spot and Dog. In the real, Spot and Dog are one. It is a strictly logical relation that holds between the abstracted concept (the predicate Dog) and the subject Spot. However, the relations “is taller than” and “is greater than” are real relations. They hold in the real whether anyone knows it or not.

Mathematical logic is therefore more like mathematics or some variety of formal science, but it isn’t logic.

• I think this definition of logic may be subject to discussion. It would be great to link those concepts (this definition of logic, intentionality etc.) to the relevant authors (Brentano, Husserl?). As is, readers don't know whether it is your personal view or a view which is supported by a school of philosophy. Mar 6 at 22:46