An Investigation of the Laws of Thought

Description

This edition includes a modern introduction and a list of suggested further reading. Before 1854 when George Boole published The Laws of Thought, the subject of 'logic' in the western world was entirely restricted to the study of the Aristotelian syllogism, which was as old as 384-322 BC. In this work, Boole introduces a sort of algebra of propositions with probability. Boole's logic forms the basis for present day Boolean Algebra, which in turn lies at the base of computer science. Because Boole is essentially inventing a number of new concepts, the discussions concerning his ideas of logic are both accessible to the non-specialist and fascinating for the historian or philosopher of mathematics and logic.

If one endeavors to give a purely formalistic answer to the question of the foundations of mathematics, then I would submit (and likewise spare a smile) that one is indeed running the risk of “underdetermining mathematics by [a strictly operative and hence restrictive view of] axioms.”

The reason being that, from a formalistic (= restrictive) understanding of axiomatics, any mathematical system (meaning, ultimately, any logical axiomatic system) is secured if it is consistent, complete, and decidable. This, as I often say, is tantamount to a computer science reduction of what mathematics is. By this view, mathematics is essentially nothing but a constructivism pretending to be “axiomatic” (which it may locally be, within the constrains of a given set of rules aiming at a given application), even though its axioms are strictly functional and thus essentially meaningless (outside of its particular rules of applicability). Thus, they don’t actually have the status of “fundamental truths,” St. Thomas’ principiis , which all sciences (as does, supremely, sacred science) are ultimately made up of (beyond the grip of any constructivism).

It is interesting that one of the pre-20th century founders of modern symbolic logic and computer programming, George Boole, still relying (like Peirce) on ontologically sound epistemological thinking, wrote the following at the beginning of his An Investigation of the Laws of Thought (as he takes the pain of distinguishing between “primary/fundamental laws and principles” and “general” yet “secondary and derived” truths):

“Let it be considered whether in any science, viewed either as a system of truth or as the foundation of a practical art, there can properly be any other test of the completeness and the fundamental character of its laws, than the completeness of its system of derived truths, and the generality of the methods which it serves to establish. Other questions may indeed present themselves. Convenience, prescription, individual preference, may urge their claims and deserve attention. But as respects the question of what constitutes science in its abstract integrity, I apprehend that no other considerations than the above are properly of any value.” (p. 4)

When Boole talks about “the laws of thought” (the ontological existence of which he holds as undeniable), he still assumes and implicitly recognizes likewise some “correspondence between the forms of thought and the actual constitution of Nature” (p. 324), while maintaining their mutual irreducibility (insofar as, if relative correspondence there somehow is, we cannot go about mistaking the map for the territory; that is, we cannot go about thinking, as Boole himself puts it, that “one system is the mere product of the other”). I think it would be more useful, in the context of his conclusive philosophical musing based on his method of expressing the logical laws of the mind using two truth values (0 and 1) and two operations (+ and ×), to view the connection between “both systems” (i.e. the intellect and Nature) by way of analogy (which, logically, can afford an extension of Boolean algebra, beyond his and Leibniz’s binary matrix).

Boole, along with De Morgan and Peirce, were closest to Leibniz’s project (of lingua characteristica universalis as “rational calculus as thought”), and even to Blessed Raymond Lully’s Ars genralis ultima (cf. my The Scholastics’ Neglected Heritage , note 2). They all endeavored to restore, expand, and establish logic as a fully formalized and mathematical science (the limit of which Gödel would eventually show, using mathematics and the laws of logic). But, I do think they all would have remained clear of the computer science reductionism we suffer from today due to an axiomatic philosophy of sheer mechanization of calculation, which seems to fit in with the gradual rise of the transhumanistic worldview.

Logos Dei, miserere nobis.