(Bolding mine)
So do rocks. Or atoms. Or galaxies. Without a comprehending observer, they’re just things.
That’s one of the drawbacks of “constructivist” arithmetic. The “succession” property of the counting numbers must be assumed to fail at some point. This kills closure under addition, which means the numbers aren’t a “group” any longer.
It takes some of the fun away from our favorite toys!
I think that is an interesting statement. I disagree. I am interested in your response.
In your POV the observer has the significance and not the object. I think the object is more important “philosophically” than the observer.
If things don’t have an intrinsic value, ie, rock vs lava vs water… how can we draw an observation if they are not fundamentally different?
Need a larger number base.
I’ve felt exactly the same way! We think of numbers like 10^10^10^10 as ridiculously big, but Graham’s number
g = F(F(F(F(F(F(F(12,3),3),3),3),3),3),3)
is preposterously bigger and utterly incomprehensible. And of course it doesn’t end there; what about
F(F(F(F(F(F(F(g,g),g),g),g),g),g),g) :eek:
By comparison, infinite numbers like ℵ[sub]0[/sub] or P(ℵ[sub]0[/sub]) seem very tame.
But infinite cardinals are not without preposterosity either. Can any of the Board’s mathematicians explain, in simple words, Woodin cardinals and why they matter?
And enough different digits to make the number base functional. There’s your problem.
Worse, it introduces new arbitrary rules about arithmetic. Somehow we must add into our use of numbers rules that place limits on their use. That there are things we can reason about in certain ways, and others that we cannot. Or at least not reason and talk about in quite the same manner. The true constructivists are perhaps not so bad, but those that insist that only numbers within human grasp are meaningful become quickly tiresome. They insist of an ill defined set of rules about what arithmetic can and cannot be done, and can only do so by adding more and more arbitrary constraints.
In the end I very much doubt you could get a coherent definition of “number” out of any of them. Not one that was not subject to redefinition each time you asked a difficult question.
Well, they are all just protons, electrons and neutrons in varying arrangements… Except, they’re not, really. Even those “fundamental particles” are just models that we use to observe and classify things, just like we do rocks, or numbers, or anything else.
I would say that existence is ultimately a relationship between observer and observed, or conception and conceiver for things like numbers that can’t be adequately represented, or objects like black holes that by their nature can’t be directly observed.
Let’s say we hypothesise that there exists an alternate universe that not only can we not observe, but no-one or nothing ever could or ever will observe it. My opinion is that talking about whether or not that universe exists is actually meaningless, as it is completely impossible to ever determine it.
As we see, OP is getting at, in part, not so much “countability” as an abstract concept, but, at least in the angle Mangetout picks up on, and which Chronos restates in admirable reductio, on countability by humans, including OP.
I cite these two because I think the are the best explanation previous to explanations of things at a more abstract level.
Having said that, I’d like to complicate Mangetout in a simple way, an idea fleshed out often in exquisite literary form by the great Jorge Louis Borges.
Mangetout mentions countability as individual sets of apples. For the sake of the argument he two "cognizables, each of one real countable-by-human (not up to nine, as noted) real things. (I am not getting into other roads to/roads of metaphysics-via-mathematics, let alone metaphysics qua metaphysics, so I don’t need quote marks around that use of “real.”) [And thanks for allowing me to use the “qua.”]
The recurring theme of Borges–in one story, eg, it is cognitive by a human of shuffles in the deck of lived experience by one man (“Funes”); in another, it not one man who who tragically remembers each; in another ("The Chinese Emperor’s Dictionary), it is the unenviable task of scholars to portray these shuffles for a book, in case the Emperor can’t remember.
Let me return to Mangetout’s apples (a happy username/post). An apple can be rotten, half rotten, poisonous, all depending on mood of the second of observation, the position of the sun at the instant of observation, that it is any or all of the above, those shuffles of whatever the human mind (again, returning to the point of the posts I cited) which may be devoid of enumeration/countability/mental cognition of sets.
Perhaps this is what Mangetout is on to. But anyway, I liked it. (Also, to all interested in OP, read Borges. He returns to this idea again and again, with bifurcation paths, the library of Babel, you name it, with exhilarating clarity and beauty, faux scholarly papers written by a poet.)
Humans impose these values/generalizations/categories, because our brains have evolved to do so (lest we get eaten by lions or whatever).
Read the Borges short story “Funes the Memorious” to get a feeling for what it’s like if we DIDN’T do this…but also to better understand that there’s no INTRINSIC reality to these values/generalizations/categories.
Addendum re Borges cite above: The book for the Chinese Emperor I mention is in his essay-“fiction” “The Analytical Language of John Wilkins,” another exploration of the themes I mentioned above, in which a language is proposed for one word for everything (the difficulties of which are particularly great for the lexicographer given the scope as suggested); the proposed book discussed as an example is called “The Celestial Emporium of Benevolent Knowledge.”
That the book is cited with the context of how such a book presents its own (insurmountable?) difficulties within a discussion of a language in which it could conceivable be written is a virtuoso demonstration of (fractals? Cantor ideas?) the very point.
The “analytical language,” as a proposal, is real: An Essay towards a Real Character and a Philosophical Language, published in 1668 (An Essay Towards a Real Character, and a Philosophical Language - Wikipedia).
ETA: or what JKellyMap said.
And, like came up earlier in the thread, which human’s grasp? A trained mathematician can easily understand large and complicated numbers, while an average person might not be able to grasp ‘how many bricks are there in the courtyard of the school’ without effort, and someone low functioning may not be able to get beyond finger counting.
“God made integers, all else is the work of man.” – Leopold Kronecker
I think one could make a case that the integers are actually the most artificial of numbers. If I have one apple, and then I get another, what’s the ratio of the new amount of apple to the old amount? It almost certainly isn’t exactly 2.000000 , since apples vary in size. As do sheep, as do rocks, as does nearly everything.
Seems like the concept of a property of a set is as artificial as a label for an object.