you can check on intuitionism versus Platonism versus formalism, etc. — Count Timothy von Icarus
model theory works with uncountable symbols — Count Timothy von Icarus
For every real there is the theorem of the form x = x. — Count Timothy von Icarus
A symbolic system with a unique symbol for every real cannot be smaller than the reals, no? — Count Timothy von Icarus
Correct, if there is a unique symbol for each real, then the set of symbols is uncountable.
You don't know what you're talking about. In ordinary model theory for ordinary first order languages, there are only countably many symbols in the language. That does not contradict that the universe of a model may be uncountable.
Everything else you responded to was you jumping over yourself to demonstrate knowledge about terminology and irrelevant. — Count Timothy von Icarus
you jumped to the formula x = x; — Count Timothy von Icarus
I'm talking about the informational encoding of any object such that a system can recognize that encoding X uniquely specifies Y without any symbolic manipulation having to be performed. As I mentioned originally, this is in the context of communications theory. — Count Timothy von Icarus
responds to a point about how some philosophers of mathematics don't think numbers exist outside formal systems, games we set up, with "they don't need to care about the philosophy of mathematics to know that 2+2 is 4," they aren't particularly interested in a discussion. — Count Timothy von Icarus
I don't think most mathematicians particularly care that much about the philosophy of mathematics.
— Count Timothy von Icarus
They don't need to care about the philosophy of mathematics to know that 2+2 is 4. — TonesInDeepFreeze
I took the tone from posts starting with:
"All aboard the crazy train,"
"No, only as you are deluded. "
"Wrong." — Count Timothy von Icarus
Generally in field with multiple subfields where the same term can refer to multiple things, it's common to ask if there might be a communication problem, not call someone an idiot. — Count Timothy von Icarus
deluded or uniformed. — Count Timothy von Icarus
when someone starts an interchange with calling you deluded — Count Timothy von Icarus
you only responded to small fractions of each post — Count Timothy von Icarus
"Correct, if there is a unique symbol for each real, then the set of symbols is uncountable."
Thank you. That's all I was saying. — Count Timothy von Icarus
For every real there is a 'theorem" in such a system of the form x = x. — Count Timothy von Icarus
If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.
Is there anything wrong with completing this tree and saying that the infinite digit RL is the Golden Ratio? — keystone
Meanwhile, among other points, I hope that at least you understand that in ordinary mathematics '=' means identity, which is to say, for any terms 'T' and 'S',
T = S
means that 'T' and 'S' name the same object, which is to say that T and S are the same object.
I said as much at the outset. — Count Timothy von Icarus
If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the abstraction? — Count Timothy von Icarus
Having 4 $20 bills and being given 4 more is not the same thing as having 10 ($200) and giving away 2 ($40). Having 5 apples and picking two more isn't the same as having 9 and throwing two in the fire. — Count Timothy von Icarus
That is, numbers exist as real abstract objects but computation is just a human language for describing their relationships. — Count Timothy von Icarus
If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P≠NP problem doesn't make sense — Count Timothy von Icarus
"informational encoding"
"system"
"can recognize"
"uniquely specifies"
"without any symbolic manipulation having to be performed"
Would you please say where I can see a glossary of that terminology as you are using it? — TonesInDeepFreeze
The word "instantiate" is related to "instance". If someone says, "Name some things that are red." you could answer, "For instance, roses are red, apples are red, blood is red." In other words, roses, apples and blood are instances of the property red. In other words, roses, apples, and blood instantiate the property red.
That's all "instantiate" means. An object x instantiates a property p if p(x). That is, x instantiates p if x has the property p, if x exhibits p, if x is an instance of p. All are ways of saying the same thing (with possibly some subtle metaphysical distinctions).
So, yes a property can be instantiated by another property. The property "is a color property" is instantiated by the property red.
In the formalist interpretation of mathematics, where "an entity is what it does,"
— Count Timothy von Icarus
Where can I read that that is a formalist interpretation? — TonesInDeepFreeze
I think you are confusing the set of all computable functions with the set of all equations. — Count Timothy von Icarus
If you tell a Turning Machine to add 2 to 2, that's different than subtracting 2 from 6, right? — Count Timothy von Icarus
However, if all arithmetic expressions that = 4 are identical with it — Count Timothy von Icarus
"1 + n = n + 1", but actually using the real numbers, not the variable? — Count Timothy von Icarus
I'm not talking about that tree in that context. I was talking about the three competing definitions of 'is a real number' and how easy or difficult it is to define the operations for real numbers based on those definitions. — TonesInDeepFreeze
so if 'is a real number' would be defined as just one particular Cauchy sequence of rationals, then which of the infinitely many should it be? — TonesInDeepFreeze
'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'. — TonesInDeepFreeze
And are you sure that every irrational number is one of the denumerable paths? And that the sequence of nodes of every denumerable path converges to an irrational number? — TonesInDeepFreeze
Aren't there denumerable paths that stay constant on a single rational number? — TonesInDeepFreeze
But 2=1.9. If your method entails that that is not the case, then I doubt that your method actually provides a complete ordered field. — TonesInDeepFreeze
That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal. — TonesInDeepFreeze
Here's what you need to provide for your SB proposal: rigorous definition...We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know. Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar. — TonesInDeepFreeze
Anyway, I am interested in the idea of SB used for defining the reals, as another poster has proposed, but I'd like to see that notion developed beyond mere handwaving. — TonesInDeepFreeze
arithmetic with real SB strings — keystone
it appears that each real number has a single path which can correspond to a sequence of rationals — keystone
'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.
— TonesInDeepFreeze
Can you rephrase this? I'm not sure what you're asking. — keystone
Potential infinity will suffice. — keystone
Every path (whether finite or infinite) leads to a different number. — keystone
an ever distant mirage — keystone
I can only take your word for it that you've satisfactorily worked out that arithmetic. Don't forget that you have to manage not just finite sequences but infinite ones too. — TonesInDeepFreeze
For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. But some people just can't grok the idea of the entries of a sequence getting arbitrarily close to a point but that point is not itself an entry in the sequence. — TonesInDeepFreeze
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.