No question about that. But that does not mean that we merely discover stuff, which is “aready out there”. We actively create brand new “worlds”, so to speak. Your proposition already includes the minds, who do the “enumeration”, and “manipulation” of numbers.
I’m delighted to find someone who regards this kind of topic as worth discussion, and is equipped to discuss it. Let us enjoy this. I put a libation in hand before replying and propose that you do the same, using your mind to create a world in which we are sitting in some form of body around a campfire in the autumn woods, imbibing a fine cognac and smoking Cuban cigars to keep the mosquitoes and women away.
Kindly allow me to soften you up by agreeing completely with your first proposition.
Then I would add, are the inventions of our minds which are original to us, original in an absolute sense? I believe that this is possible, yet my experience says otherwise. Upon first developing some of the ideas which I’ve expressed on CAF, I did nothing with them, figuring that someone else had long ago come to the identical conclusions and published them. They seemed so obvious and logical to me that I could not imagine otherwise. Yet I’ve not found them anywhere.
(I am certain that if I publish them to some measure of success, the woodwork will explode with others claiming my notions for themselves. If my presentation fails, we won’t hear a peep, twitter, or scraping of cockroach legs from the same smarmy lot.)
History shows us many examples of duplicate complex ideas— Newton and Leibniz both discovering calculus, Darwin and Wallace getting a similar handle on critter changes.
This raises the arcane question, where is the stuff we discover? The automobile exists in physical space, but in mental space, the Egyptians or some other old culture invented it. We simply added an extra pair of wheels, moved the horses under a cabinet at the front, replaced the reins with steering gear, and added seats.
I now believe that it is possible for any mind to invent a concept which has not been previously invented by another mind. This appears to be true if the number of possible concepts is infinite, as I suspect. Yet I don’t think that there are answers to the question, but it certainly raises other questions.
For example, I believe in a Creator, and as a consequence that this Creator is either directly or indirectly responsible for the biological machine through which I explore and interpret the universe, the planet providing its support system, the atoms from which it is constructed.
Like many others I’d like to learn more about the workings of these things, but rather than take the scientific approach of trying to figure them out by ripping them apart, I believe that a better approach is to understand their workings by reinventing them. This would involve, in effect, the discovery of something which already exists in mind-space.
And, since the Creator in Whom I believe is not omniscient, I can, and have adopted the conclusion that even I can think of things which He, She, It, or They have not yet thought of.
This is not a big deal. I assume, with one exception, that most of the innovations I might discover are there for discovery only because greater minds neglected them.
Of course I agree that the property of “enumerable” exists objectively. But the numbers to do the “enumeration” are “created” by the mind. In mathematics the theory of prime numbers is one of the most beautiful concepts. There is the theorem, which says that every positive integer can be factorized by multiplying its factors (prime factors) in one and only one way, disregarding the order of the factors (for example 6 = 2 * 3). An awesomely beautiful theorem… but is it really “universally true”? No, it is not. If those hypothetical mathematicians from the Arcturus would happen to use a different notational system, like defining integers as “a + b * sqrt(-5)”, then the prime number theorem does not hold any more. In that system 6 would be either “2 * 3” or (1 + sqrt(-5))*(1 - sqrt(-5)), two different factorizations.
While we commonly speak of
the theory of prime numbers, do we not really mean,
The Theory of Prime Integers? Of course I would not expect a theory of integers to apply to fractions, complex numbers, or vectors.
My sense of things is far more basic. I believe that minds naturally discover certain basic mathematical concepts. So far as I know, even primitive societies of men know how to count. I also believe that the same concepts we are so proud of discovering have been found by more advanced beings, long ago because they are available for discovery.
(Submitted w/o edits to beat the coming power outage.)