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June 7

Ultrafinitism and strict constructivism

I note that ultrafinitists and strict constructivists deny the existence of numbers like floor(e^e^e^{79}) as being only formal arrangements of symbols, but seem happy to consider numbers such as a billion as being real, because they can easily be calculated using simple arithmetic, counted up to using an electronic counter, or physically constructed using current technology (for example, a billion transistors on a single silicon die). But are numbers like a billion actually comprehensible by the mathematically unaided human mind? I can't subitize any number bigger than about six without thinking of it as a pattern (and thus using symbolic thinking), and I very much doubt whether anyone has ever counted to a billion, or could take in the sight of a billion objects and grasp the idea of "billon" from it, instead of just "uncountably many".

~~Which leads me to my question: is there a "human scale" version of "primitive" constructivism that disdains all symbolic and mechanical aids, and if so, what is it called?~~

For an example of what I mean, say I regard only the numbers one to six (and possibly zero) as my set of basic objects, and then allow myself only to make constructions on top of them using only simple constructs (adding, multiplying, taking away) that involve no more than six steps? If so, how far can we go? Can we talk about "doing mathematics" in this very small world, and would mathematics look like, if we did? There will obviously be numbers that link back to the whole in many different ways, but there will also presumably be "border" numbers that can only be reached in one way -- and beyond these, would there be "dark integers" in Greg Egan's sense?

Now here's the interesting thing. People with better aptitude with visualization and abstraction would actually have different mathematical worlds from others, with those with lesser capabilities denying the reality of the mathematical worlds of those with greater abilities. If so, would this really any different from debates like finitism vs. non-finitism? -- The Anome (talk) 13:23, 7 June 2015 (UTC)[reply]

Update: I've just found http://www.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/Shell_1st-Maloney-constructivists.pdf , which seems to deal with everything I was thinking about in my question. -- The Anome (talk) 13:53, 7 June 2015 (UTC)[reply]

Resolved

StuRat (talk) 18:47, 7 June 2015 (UTC)[reply]

What are formalizations without rigor and rigorous formalizations?

What are formalizations without rigor and formalizations with rigor? And what are some examples of each one? How can you tell a formalization has no rigor? Is rigor only another word for incomplete?--Llaanngg (talk) 15:15, 7 June 2015 (UTC)[reply]

I'm not clear on the context here; can you give an example where the phrase "formalization with/without rigor" is used? To me, formalization is where you divorce the terms and symbols of a mathematical theory from their original meaning. For example a formal power series is where you keep the notation of a power series but make it independent of the the idea of a power series of a function. Rigorous, as opposed to heuristic, is something that applies to proofs. A heuristic proof is one which is incomplete but which argues that the statement ought to be true and which might be turned into a rigorous proof. In contrast, a rigorous proof has no gaps and relies solely on the rules of logic. I think formalization is largely a matter of creating definitions, and definitions aren't subject to proof or disproof. Definitions can be vague or ambiguous though. --RDBury (talk) 13:33, 8 June 2015 (UTC)[reply]

No, rigor is not a word for incomplete. I think I understand the questions and confusion: because of the way 'formal' is used to mean different things, it often gets conflated with rigor. Often the two go together: One way to make a proof more rigorous (good, complete, unassailable) is to take the time and care to write up a proof with well-defined terms and symbols (the "forms"). This is discussed a bit at Rigour#Mathematical_proof, and the example that Cauchy gave rigorous foundations to works of Gauss and Euler is apt.

However, not everything formal is rigorous. Sometimes you just want to manipulate the symbols and see where that takes you, without caring much a priori about the ontological nature of the objects you're playing with. That is the idea behind the formal derivative and the formal power series (confusingly enough, these things can be dealt with rigorously, but that is not the historical path or the way they are commonly taught.) As for rigor without formality, that's a little harder to come by, but I suggest there are good candidates among the proofs without words - they are rigorous in that, if the reader understands all the implicit assumptions and words that could go with them, then the reasoning is both logically sound and logically valid, and we can conclude that the claim has been rigorously proven, despite the fact that we don't have careful definitions, symbols, etc of formality.

Finally, there is no universal way to tell if a formalism lacks rigor. There may be a few "tells" or indicators of lack of rigor, similar to how we can (sometimes) identify a crackpot or conspiracy theorist by their tone. But in the general sense, there are no shortcuts, each case must be examined in detail by, preferably by experts in the field with ample time. For example, it took a while for everybody to agree that Grigori_Perelman had really proven the Poincare conjecture - lots of people had to read/review it to make sure it was correct, and it's probably fair to say the proof wasn't made fully rigorous until other authors read the work and published further details. Even here on the ref desks, we occasionally get someone claiming they can trisect an angle, and it can be very difficult to go through and spot the specific error in reasoning, even though we know the claim is false. SemanticMantis (talk) 15:23, 8 June 2015 (UTC)[reply]