Who's Korzybski? A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
I think the main implication of Incompleteness is that some truths are axiomatic and based in (horror...) subjective experience. They can't deduced because they're primary qualia.
If put two similar things next to each other, you experience the twoness of the similar things rather than eighteenness similar things. (Ceteris paribus, etc.) The experience of numberness and of basic addition and subtraction is probably innate, although some cultures develop it much farther than others, while others barely develop it at all.
Some animals can count too, after a fashion.
Subitizing, coined in 1949 by E.L. Kaufman et al.[1] refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus (meaning sudden) and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range.[1] Number judgments for larger set-sizes were referred to either as counting or estimating, depending on the number of elements present within the display, and the time given to observers in which to respond (i.e., estimation occurs if insufficient time is available for observers to accurately count all the items present). The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid[2], accurate[3] and confident.[4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence.[1] In addition, response times rise in a dramatic fashion, with an extra 250 ms - 350 ms added for each additional item within the display beyond about four. ... So, while there may be no span of apprehension, there appear to be real differences in the ways in which a small number of elements is processed by the visual system (i.e., approximately < 4 items), compared with larger numbers of elements (i.e., approximately > 4 items). Recent findings [7] demonstrated that subitizing and counting are not restricted to visual perception, but also extend to tactile perception (when observers had to name the number of stimulated fingertips).
The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid[2], accurate[3] and confident.[4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence.[1] In addition, response times rise in a dramatic fashion, with an extra 250 ms - 350 ms added for each additional item within the display beyond about four.
... So, while there may be no span of apprehension, there appear to be real differences in the ways in which a small number of elements is processed by the visual system (i.e., approximately < 4 items), compared with larger numbers of elements (i.e., approximately > 4 items). Recent findings [7] demonstrated that subitizing and counting are not restricted to visual perception, but also extend to tactile perception (when observers had to name the number of stimulated fingertips).
But is this qualia? Applying sequential attention within the subitizing range doesn't lead to different results. A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
But is this qualia?
No.
Qualia is the mental state of "knowing what it is like" to have a particular mental state.
Axioms can't be deduced because they're axiomatic. A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
Also, thinking again about subitizing, it is not only repeatable but also people can agree on the result. The qualia (what it feels like to subitize four as opposed to counting to four) is irrelevant to a large extent. It's just extremely interesting that we can subitize and it may even have linguistic implications for grammatical number, but just because your native language doesn't have counting numbers doesn't mean you can't learn them, and the possible connection between perception and grammar is not so surprising since both perception and grammar involve the same brain. A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
I'd be surprised if basic arithmetic - certainly addition, possibly subtraction, probably not anything more advanced - wasn't similarly rooted in experience.
But the basic point was that arithmetic is rooted in experience, and doesn't exist independently of it. Trying to prove it using formal logic makes for an interesting scenic trip, but eventually you end up standing over a hole which logic can't fill for you.
Axioms - to use the term strictly - establish the Rules of the Deductive Game you're playing, intellectually speaking. Nothing prevents anyone else from declaring other axioms and playing other games. This is exactly how non-Euclidean geometries is/are developed.
Deductive Truth arises from the manipulation of the Terms and Operations, within the Axioms, of A Deductive System - as guided by the particular Interpretation of that system. Even within the good old Categorical Logic there are two Interpretations: Aristotelean and Boolean.
Verification of the results - the Truthyness of the Truth Value ;-) of a particular Axiomatic System - is a whole 'nuther topic.
Who's Korzybski?
He created General Semantics. I mentioned him several times here. "Dieu se rit des hommes qui se plaignent des conséquences alors qu'ils en chérissent les causes" Jacques-Bénigne Bossuet
