Mathematics without Infinity
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 refining this idea?
What exists, what is real, is what’s exquisite, for the very fact that it’s not merely an imaginary figment. An imagined “perfect circle” is less perfect for not having that one quality: existence. “Perfect circles don’t exist!” So what makes them so perfect then?
That’s a “flipped classroom view” from the standpoint of many a metaphysician, as the more conventional claim is that “true perfection” is privately witnessed only in some private beholder’s mental eye. The empirical, phenomenal “real world” is considered fallen, a sinful place, next to which we enjoy private access to an introspected world of Platonic purity, a space of “perfect polyhedrons” and other such beetles, a secret garden.
In the energetic world, precision comes at a cost. Controlling phenomena to an ultra micro scale takes a lot of engineering and machinery. Chip fabrication. Chemical laboratories. Nanotechnology. We keep pushing the limits.
Mathematicians, in contrast, seem to get all the precision they need just by closing their eyes and imagining the exact point that is signified by some Greek letter. They pity the engineers, who battle with thermodynamics and quantum mechanics to achieve any kind of control at the nano level.
Mathematics provides a friction free environment, a comfy sandbox for the unfettered imagination.
The Finitist does not require “planes of infinite extent” nor infinitely minute points, so small that atoms seem the size of Jupiter in comparison.
Why let infinity intrude. Why get all hand wavy? Choose a scale to work on and admit that some points are thereby too small to tune in. They’re “infra-tunable” i.e. beyond the scope of the currently coordinating system.
Speaking of infra-tunable points, a second influence tilting me towards Finitism, besides Wittgenstein, , would be R. Buckminster Fuller and his mathematical philosophy.
His magnum opus, Synergetics, is definitely a contribution to the lineage of Democritus (discrete mathematics), more so than to that of Euclid (continuous mathematics).
Quoting from said work of prose:
1001.25 The misconception of a “straight” line and its popular adoption into humanity’s education system as constituting the “shortest distance between two points” takes no consideration of what the invisible, dynamic, atomically structured system may be which provides the only superficially flat paper-and-lead-pencil-pattern of interrelationship graphing of the line running between the two points considered. Nor does the straight-line shortest-distance assumption consider what a “point” is and where it begins and ends — ergo, it cannot determine where and when its dimensionless points have been reached, and it cannot determine what the exact length of that shortest distance between “points” may be.
1001.26 Such self-deceiving misinterpretations of experiences have been introduced by education into human sensing and traditional reasoning only because of humanity’s microstature and microlongevity in respect to the terrestrial environment and geological time. Individual humans have also been overwhelmed by the momentum of tradition, the persuasions of “common sense,” and a general fear of questioning long-established and ultimately power-backed authority and tradition. Thus has innocent humanity been misinformed or underinformed by the spoken-word-relayed inventory of only popularly explained, naked-eye impressions of local environment experiences as they have occurred throughout millions of years prior to humanity’s discovery and development of instrumentally accommodated, macroscoped and microscoped exploration of our comprehensive environment. The experientially obtained, macro-micro, instrumentally measured data found no evidence of the existence of dimensionless “points,” “lines,” and “planes,” nor of dimensioned “solids,” nor of any “thing,” nor of any noun-designatable, thing-substantiated, static entities. The experiments of human scientists have disclosed only verb-describable events — four-dimensionally coordinate behaviors of complexedly and ceaselessly intertransforming events, wavilinear event trajectories, interferences, and resonant event fields.
Once we see that the meaning of infinity comes from shared language games in the public arena, and not from privately introspected qualia, we’re in a better position to get free of its adolescent appeal.
Mature adults need not treasure infinity as the bees knees, in any and every mathematical system.
There’s room for Finitist philosophies on Planet Earth.