Thanks for engaging with what I've previously asked!
To go back to Bateson's initial quote, what would a numberless measurement of length, for example, be? - javra
Couldn't this be accomplished by simply referencing objects' extension in relation to one another? — Count Timothy von Icarus
Yes, you are quite right. (Realized this after posting, but deemed that editing it would be a bit much ... in any case, my bad here.) I would however better reword the concern I have in this way:
-- Can one have any measurement that is devoid of
any discrete givens which we - either thinkingly or unthinkingly - enumerate (i.e., determine the amount of) via numbers?
If not, then it currently seems to me that measurement necessitates number in some way or another.
Notice that in the very quoted sentence number (a definite amount; i.e., a definite quantity) is necessarily specified in order for cogent semantics to obtain (what I've boldfaced). Likewise, while a ratio might not be itself interpreted as a number (debatable) it will yet, I so far find, necessarily consist of a relation between numbers - at the very least between quantities (the plurality of which is itself a quantity) which we hold the potential to enumerate. Else, in measurements that strictly concern relations, such as greater than or lesser than, there will always be an at the very least implicitly addressed number of givens to which the relations applies. I'm for example weak on pure theoretical mathematics, but I so far can't find any exception to this.
I feel like there is support for the supposition that the illusion of discreteness is just a useful survival trick as much as for the idea that innate numeracy denotes the existence of numbers "out there, sans mind." — Count Timothy von Icarus
As to whether lesser animals can count, as philosophy it's right up there with whether lesser animals are in fact conscious - to which might as well be appended the issue of other minds. In short, I'm convinced that they do, but, as with those who'd disagree, can't provide conclusive philosophical evidence of it - at least not in a forum format. So, I won't debate the issue.
Still, the pivotal issue I was addressing is that, as I currently find it, discreteness is contingent on the occurrence of awareness - such that if awareness then discreteness (and as an important meta-example: the occurrence of one awareness or more will each be a discrete given). And, furthermore, that numbers are only then contingent on the occurrence of discreteness. This irrespective of one's metaphysical interpretation(s) regarding the consequent significance in respect to the cosmos we inhabit. (e.g., a materialists' view that an awareness-devoid cosmos is possible or, else, an idealist's view that such is impossible - as two among other metaphysical perspectives)
So I'm here in full agreement that "numbers 'out there, sans minds' [by which I here understand, tersely stated, "a plurality of discrete awareness"]'" can only be a fallacy.
how can one have numbers in the complete absence of discrete amounts of givens - i.e., of quantities? - javra
Imagine a continuum, for example a line, of finite length. Our line has an uncountably infinite number of points but also a finite length. — Count Timothy von Icarus
Right here, in the very semantics of what a line is, is the occurrence of quantity in the form of "points" - such that this quantity minimally consists of more than one point. In addressing "a" continuum one is likewise specifying a quantity - not two or three continuums but one. So the occurrence of quantity is a requite aspect of any continuum - be it real or strictly conceptual.
ake some section of the line, arbitrarily, and compare how many lengths of the section fit within the whole. There are sections of the line that exist such that the line can be broken into n segments of equal length, where n is a natural number. No initial discreteness required, right? All that is required is that the points of the line differ from each other in some way; — Count Timothy von Icarus
Maybe the "discreteness" here addressed by you has a specialized mathematical meaning? But in the ordinary sense I've addressed it it specifies something being separate, distinct, individual. Hence, in the sense I intend the very sections of the line that are compared are thereby discrete (to our awareness of them as such - otherwise no comparison could be made).
BTW, one could then address point-free topology as another example to be provided - but, here too, tmk there will be discerned some form of separateness somewhere (e.g., sets), such that discreteness (and hence quantity) yet obtains.
So, I again find that the (maybe I should specify, cosmic) occurrence of quantity is requisite for the occurrence of numbers (be the latter's occurrence also cosmic or, else, strictly located in individual minds as some would have it).
I've always found the reverse argument more interesting, the claim that numbers are essential for reality, or at least our understanding of it. — Count Timothy von Icarus
This in fact isn't too different from my own personal metaphysical views. Only that I maintain quantity - as in "discrete givens" - to be essential to existence rather than to reality (with "existence" here roughly understood as all which "stands out" in any way) and, thereby, to physical reality (which exists); this, thereby, concurrently necessitating the ontic occurence of numbers in the cosmos (however sentience might represent them symbolically; e.g. as "IV", as "4", or as "four"). But I don't want to digress into my own metaphysical views concerning this.
All the same, of main interest here is the issue of how numbers could be had in
the complete absence of quantity.
Anyhow, if some hitherto unformulated version of logicalism is true, and numbers are reducible to logos, it seems to me like this argument is moot (and that the concept of logos spermatikos ends up beating out divine nous as a better explanation of "how things are," IMHO.) — Count Timothy von Icarus
:grin: I could go with that. (but then these touchy terms hold different connotations to different people, for instance, that of "spermatikos" say by compassion to the terms "in-fluence" or even that of "inspiration (aka, to breath in or, more archaically, roughly, to be breathed into psychically)") But yes, a Heraclitean-like, cosmic logos of the type addressed stands in direct logical contradiction to an omnipotent and omniscient creator deity whose "words" make up the world.