For the philosophically unsophisticated, why is it significant that logic cannot

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For the philosophically unsophisticated, why is it significant that logic cannot be reduced to mathematics? What difference would it have made if that project had succeeded; what is import that it failed?

Your ability to balance your checkbook, or to draw logical inferences in everyday life, won’t be affected in the least by difficulties in figuring out just how logic and higher mathematics are connected. Nevertheless, the relationship between logic and mathematics has been an intriguing conundrum for the better part of two centuries.

There have been many attempts to understand various aspects of logic mathematically, and perhaps the most famous is George Boole’s Mathematical Analysis of Logic (1847), which laid the foundation for Boolean algebra. Far from being a failure, Boole’s effort seems to have been a smashing success, especially when we consider the extent to which Boolean algebra underlies modern digital computing.

Nevertheless, the relationship between logic and mathematics can go in two directions, not just one, and so, just as one might try to understand various parts of logic mathematically, one can also try to understand various parts of mathematics logically. It is this further possibility, I suspect, that has prompted your question about a “failure.”

Late in the nineteenth century, the German logician Gottlob Frege sought to understand part of mathematics in terms of logic. Frege wanted to reduce arithmetic to logic, and later writers tried to reduce other parts of mathematics to logic too. Today, this approach is usually called “logicism,” and the primary motivation behind it is to discover exactly what kinds of entities mathematical objects are.

When we do arithmetic, for example, we add numbers, but what exactly is a number? Is it a physical object? Is it just an idea in our heads? Is it a mere symbol? Is it a timeless, placeless eternal entity that exists even if no one thinks about it? These questions about numbers are as old as Plato (maybe older), and they generally fall under the heading of ontology—which asks what kinds of objects exist. Logicism is an attempt to answer these ontological questions, and this is why it seeks to “reduce” mathematics to logic.

In 1931, Kurt Gödel demonstrated that no logical system rich enough to include arithmetic as a consequence could be shown within that system to be both consistent and complete. Either some statements of the system would have to remain unprovable, or if provable, the system would be inconsistent. Many have argued that Gödel’s result showed that logicism must fail, or at least that some versions of it must fail, but it is important to add that the exact impact of his result on logicism is a complicated question, and subject to different interpretations. However this may be, all these discussions concern logic and mathematics as expressed through formalized symbolic systems, which were developed in the late nineteenth century, and in the twentieth century, and these discussions have had, in fact, no real effect on our ordinary reasoning in daily life, or on our everyday ability to add and subtract correctly.

Lest these last remarks seem philosophically controversial, let me say a bit more to explain them.

In Isaac Newton’s day, none of these formalized systems—systems of mathematical logic or of fully symbolic logic—existed, and yet hardly anyone would say, I think, that without these systems Newton was unable to add simple sums correctly or to draw logical inferences correctly. It follows that his ability to do these things was quite independent of such systems. More broadly, formal systems of logic and mathematics can certainly improve and refine our logical and mathematical abilities, but the abilities are already partially present in us without the systems, and it is precisely because we have some of these abilities antecedently that the formal systems can be constructed at all.

These same logical and mathematical abilities are also antecedent to our musings about ontology, or to our disagreements about ontology. Historically, there have been many different theories of what numbers are, just as there have been many different theories of what kinds of entities the propositions of logical argumentation are. Nevertheless, two and three have always made five, and the Barbara syllogism, at least in ordinary cases, has always been valid. Ontology is still a fascinating subject, to be sure, but its practical effects are often quite limited. Ontology certainly does affect higher mathematics, but there are large stretches of ordinary reasoning and arithmetical reckoning that are essentially immune to it.

So whatever results might be derived from various oddities in the further reaches of logic and mathematics, you can still be tolerably sure that, if all cats are cool, and Felix is a cat, then Felix is cool. And you can be equally sure that, if you have five rabbits, and you then add seven rabbits, you will have twelve rabbits, at least for a while.

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