To the Infinite from the Finite

Infinity, the concept, has proven to be a vexing one:

1. Greek thinkers apparently considered notions of infinity to be abhorrent. For example, Socrates himself is alleged "to [have] reject[ed] the idea that there can be different sizes of infinity."
2. It took the better part of a half-century to develop modern techniques like Dedekind Cuts, Diagonalization, well-behaving Ordinals, and modern number taxonomies (the Reals, Integers, Floating-Point Numbers, Imaginary, Complex, etc.).
3. Ditto for Cardinality, Countable and Countably Finite Sets, and reframing Infinity as a problem of Sets (rather than with concept of Numbers or as geometric paradoxes).

Technical worries (from within the field of mathematics) aside, a number of philosophical concerns have been raised over several centuries. (And, often proved surprisingly influential in the development of the techniques above. Cantor being heavily influenced by philosophy is often touted as a paradigmatic example.)

One line of criticism that I've alluded to before (Page 25) stems from the ancient worry formulated along the following lines of criticism:

1. [EPISTEMIC]
   • How could finite minds creatively construct the concept of infinitude?
   • The infinite cannot be derived from the finite.
   • It must therefore be an objective, discovered, reality.
2. [LINGUSTIC]
   • If an infinitude is an infinite sequence,
   • And, all number systems are symbolic / written.
   • How could we even come to the concept of an infinitude in the first place?
   • Therefore, number systems aren't purely symbolic / written, they are discovered.

The latter concern stems from arguments against Constructive Mathematics. The former arises in various theological contexts.

Previously, I alluded to a potential line of reply to the above that I think is interesting and original. The hypothesis goes something like the following:

1. Begin with finitude as a concept. (No one contests that finitude is something finite beings can be cognizant of, etc.)
2. Define a predicate to represent that concept.
3. Arrive at infinitude by negating the predicate.

Hence, the infinite can straightforwardly be constructed (even to the standards of Constructive Mathematics at least many of the common variants - which I think is the most intriguing takeaway and insight into the above that I pointed out) from finite concepts. Negation is the forgotten mechanism (on such a view).

I credit Dr. Richard Han with the overlooked and brilliant line of thinking above.

Very few thinkers I'm aware of have attempted to arrive at the infinite from the finite (most seeming to accept the intuitive pull and conclusion of the above concerns) - almost all thinking seems focused on criticizing infinity as a stable or legitimate notion and then calling for a general retreat "backward" to the finite.

Turns out computer science has also taken a similar approach. (Yet another example of how much academic philosophy would benefit by more carefully following the happenings of other disciplines. Though to be fair the specific concerns [EPISTEMIC], [LINGUISTIC] above are a fairly niche subdiscussion in the broader debate about infinity which is neatly summarized in the great SEP article.)

Computer Science

I was unaware that linguistic treatments of Infinity have been a topic of concern (for the broader computer science community) since at least 1985 with the famous IEEE-754 and IEC-559 standards that define Floating Point Numbers and establish certain conventions around NaN and Infinity.

These, in turn, informed the C++ libraries: std::numeric_limits::has_infinity and isinf which take a similar approach: they define a predicate (truth-functional boolean method) to determine whether an expression is an infinity.

There are some key differences:

1. Most computer programming languages stipulate the existence of an infinity symbol: ∞ (Unicode symbol U+221E being widely accepted - some languages use Infinity).
2. The programming languages then build up several methods to check for those values.

So, the approaches of the software engineering community parallel some of the ideas above but assume the existence of infinity in the first place.

Constructive Mathematics

There are several fundamental points of disagreement between Classical Mathematicians and proponents of various Constructive philosophies of math (Intuitionism, Formalism):

1. The validity of the Law of Excluded Middle (and Bivalance, Double Negation Elimination, thereby).
2. The validity of Reductio ad Absurdum
3. Constraints on Mathematical Induction
4. Constraints on Mathematical Ontology (are there infinities, Constructive Proof)
5. Disagreements about the legitimacy Cantorian Diagonalization-style proofs
6. etc.

An astute observer will note that this mostly from worries like [EPISTEMIC] and [LINGUSTIC] (e.g. - Constructive Mathematics is motivated by skepticism against Cantorian-style Infinities which leads to rejecting many commonly-accepted techniques in favor of a restricted subset of mathematics.).

A curious thing if finitude and infinitude weren't as problematic after all. On the one hand, we arrive at infinitude on the basis of finite proof and using entirely finite concepts. (Arriving at infinity.) On the other hand, we have constructed a new mathematical predicate. Clearly, the broad concept of "infinitude" doesn't give us the more-grained (and useful) notions like Transfinite Cardinalities (of specific kinds) but it may show that getting "to infinity" may not be that "beyond" the finite after all.