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Mathematics without Infinity

Kirby Urner
5 min readMay 18, 2018


The circle of folks willing to philosophize out loud, about matters mathematical (hello Gilbert & Sullivan), is usually quite small, as the risk of making a fool of oneself are high, whereas pure boredom seems likewise a “hard place” to get unstuck from.

A mathematics without infinity, of which we could have a great many, at least in theory, might seem like double trouble insofar as “infinity” is usually seen as a draw.

People flock to the carnival to be amused by the vertigo. There’s nothing like “infinite recursion” to give a stomach butterflies, and isn’t that what they pay for?

My sparring partner, likewise a good friend, blames Wittgenstein’s influence for my pronounced case of Finitism. I’ll gladly concur.

What a lot of people miss about Ludwig’s critique of the “private ostensive definition” is that “infinity” is often on the other end of that private pointing stick.

In other words, to assure one’s self that “infinity” has meaning, a solipsist might “look within” and concentrate really hard, on “that which is infinite” (privately shown).

One needs only a moment, the thinking goes, to reassure one’s self: it’s still there, that treasure box experience that is “the meaning” of infinity.

Wittgenstein referred to this kind of activity as the “beetle in the box” — begging the question, what do we really mean by “beetle”?

In Philosophical Investigations, his objective is to counter the temptation to imagine meanings as fleetingly ghostly private objects of experience. The meaning of “infinity” is not a matter of self-exposure to private phenomena. Words gain traction through shared language games. Public investigation is both possible and necessary, if “the meaning of” is what one is after.

What might be among the hallmarks of a Finitist?

In abandoning any need for “infinity” what occurs?

How is one’s thinking affected?

I’d say a sense that a “good approximation” is “good enough” is one characteristic of Finitism. We don’t require “infinite precision” regarding any specific measure. The very idea of “infinitely precise” becomes a topic for mystics and navel gazers. Where are you looking, when…