What a Number Is (1991)
csus.edu
csus.edu
>It is easy to despair as Wittgenstein did, that science and all philosophies are ultimately closed systems of circular truths.
That doesn't seem at all right to me. Apart from the question of which Wittgenstein you're talking about (early, later, possibly mid), I don't think he despaired about either of these things. He seems to have felt that philosophy consists of nothing but various confusions—hardly what our sentence claims—of which it was his and philosophy's job to clear up once and for all and leave behind. But he spent his life vigorously working away at many of them. Often happily and very productively. What despair he felt I think had other sources; he had a very strong ethical/mystical (for want of a better word) sense; also frustration at his own and others' lack of ability to think better.
Involving science in that sentence makes no sense at all to me. He was involved with the Vienna Circle logical positivists, hardly despairers about science. Later he often advised his philosophy students to give up philosophy and do something useful to society, like medicine or labouring work, anything. I don't remember any negativity towards science from him, except reminders that some questions aren't scientific questions.
One could take this further and claim that even the most rudimentary mathematical construct, the integer, is in the eye of the beholder. Counting relies on certain perceive regularities: "boundaries" that allow us determine where a the "identity" of an object, and "similarities" that allow us to group together objects into counts.
The concept of an integer seems to presuppose the concepts of identity (or distinctness) and regularity (for want of a better word). Whether all types of neurologies will have the same ideas of identity and regularities (or objects and classes?) and therefore the same concept of integers, is something that we probably could not imagine because we can't step out of our own neurologies...
I prefer to believe the universe exists and whatever is possible is synonymous with being physical law. Conscious subjective experience is one of those phenomena not excluded by physical law.
But then this article claims and you also seems to claim something like:
> But what if somewhere else some consciousness has intuitive experiences about numbers which differ from our intuitive experiences.
I don't see why minds with intuitions creates any burden for numbers. From my perspective Mathematics is at bottom a game of string manipulation. Interpretations of those strings often can be applied to understand/predict/manipulate the physical universe. But the only real thing is the universe itself not the subjective interpretations in the minds.
The minds are real, obviously, they exist in the universe. Integers are an interpretation of a particular string. Suppose some other mind is incapable of an intuitive understanding an interpretation that we as humans have; I don't see how the "integers" could possibly be affected.
None of us are concerned that dogs don't understand math. Why we be concerned about the minds of aliens and what they find intuitive?
Although at the ground level the universe appears to be quantized (at least below the Planck limit, at our macro scale it's mostly continuous. In fact in most things, if not all real world things, integers live only in some platonic universe. Perhaps we do a disservice to children by starting them with discrete arithmetic.
> From my perspective Mathematics is at bottom a game of string manipulation.
I think this really can only be true of basic arithmetic. Even in arithmetic we can reason about strings too long to be constructed in your our universe.
> None of us are concerned that dogs don't understand math. Why we be concerned about the minds of aliens and what they find intuitive?
They don't? Leaping to catch a frisbee is pretty advanced mathematics. I do doubt that dogs understand integers. But these fundamental questions about alien species whether dogs or some extraterrestrial intelligence are pretty important epistemological inquiries.
Maybe the Martin Gardiner book on the inherent polarity of the universe provides the basis for a categorical One: lets all align with our head|antenna|sense-unit|orifice-for-feeding pointing upward..