I would say that I've made a considerable effort the past several years to understand your point of view. — fishfry
When you pooh-poohed the 13-digit accuracy of the measurement of the magnetic moment of the electron, you indicated a dismissal of all experimental science. — fishfry
Luke just seems to be always looking for the easiest ways (mostly fallacious) of making me appear to be wrong, no matter what I say. — Metaphysician Undercover
There is really no reason to attempt to count the natural numbers, when we know that this is impossible because they are infinite. And numbers are not even countable objects in the first place, they are imaginary, so such a count, counting imaginary things, is a false count. Therefore natural numbers ought not be thought of as countable. — Metaphysician Undercover
Yes, I've apprehend this, and I respect it. I know that's why you keep on engaging me. it's not easy to understand unorthodox and unconventional ways of thinking like mine though, so I've seen your frustration. But I do appreciate the effort. I've see the same effort to understand from jgill. I don't think TonesInDeepFreeze quite has that attitude though, and Luke just seems to be always looking for the easiest ways (mostly fallacious) of making me appear to be wrong, no matter what I say. — Metaphysician Undercover
Let me tell you something. The magnetic moment of an electron is a defining feature of how magnetism effects a massive object. Therefore it is not measured it is a stipulation based in specific assumptions such as a circular orbit. But if the electron's orbit is really not circular, then the stipulated number is incorrect. — Metaphysician Undercover
You might think, that "the meaning is always clear from context", but if you go back and reread TIDF's discussion of counting a quantity, you'll see the equivocation with order.
All I saw in you demonstration was a spatial ordering of symbols. I really do not see how to derive a purely abstract order from this. If you truly think that there is some type of order which is intelligible without any spatial or temporal reference, you need to do a better job demonstrating and explaining it.
I assure you, I am very interested to see this demonstration, because I've been looking for such a thing for a long time, because it would justify a pure form of "a priori". Of course, I'll be very harsh in my criticism because I used to believe in the pure a priori years ago, but when such a believe could not ever be justified I've since changed my mind. To persuade me back, would require what I would apprehend your demonstration as a faultless proof. — Metaphysician Undercover
There is an issue though, that I'll warn you of. Any such demonstration which you can make, will be an empirical demonstration, using symbols to represent the abstract. — Metaphysician Undercover
So the onus will be on you, — Metaphysician Undercover
to demonstrate how the proposed "purely abstract order" could exist without the use of the empirical symbols, — Metaphysician Undercover
or else to show that the empirical symbols could exist in some sort of order which is grounded or understood neither through temporal nor spatial ideas. — Metaphysician Undercover
I'll tell you something else though, I have opted for a sort of compromise to this problem of justifying the pure a priori, by concluding that time itself is non-empirical, thus justifying the temporal order of first, second, third, etc., as purely a priori. — Metaphysician Undercover
However, this requires that I divorce myself from the conventional idea of time which sees time as derived from spatial change. Instead, we need to see time as required, necessary for spatial change, and this places the passing of time as prior to all spatial existence. — Metaphysician Undercover
This is why I said what I did about modern physics, this position is completely incompatible with the representation of time employed in physics. In conceiving of time in this way we have the means for a sort of compromised pure a priori order. It is compromised because it divides "experience" into two parts, associated with the internal and external intuitions. The internal being the intuition of time, must be separated from "experience" to maintain the status of "a priori", free from experience, for the temporal order. So it's a compromised pure a priori. — Metaphysician Undercover
I didn't deny the distinction between quantity and order, I emphasized it to accuse Tones of equivocation between the two in his representation of a count as bijection. — Metaphysician Undercover
That is exactly why I attack the principles of mathematics as faulty. There are empirical principles based in the law of identity, by which a physical, and sensible object is designated as an individual unit, a distinct particular, which can be counted as one discrete entity. There are no such principles for imaginary things. — Metaphysician Undercover
Imaginary things have vague and fuzzy boundaries as evidenced from the sorites paradox. so the fact that "there is no mathematical difference between counting abstract or imaginary objects...and counting rocks", is evidence of faulty mathematics. — Metaphysician Undercover
As I said, all you've given me is a representation of a spatial ordering of symbols. If you are presenting me with something more than this you'll have to provide me with a better demonstration. — Metaphysician Undercover
I go both ways on this. Of space and time, one is continuous, the other discrete. But this is another reason why I think physics has a faulty representation of space and time, they tend to class the two together, as both either one or the other. — Metaphysician Undercover
I don't think TonesInDeepFreeze quite has that attitude though — Metaphysician Undercover
As Wiles said when he proved Fermat's last theorem at a conference: "I think I'll stop now." — fishfry
You then stated that "we can only count representations of the imaginary things, which exist as symbols." — Luke
But I can't agree with your apparent extrapolation from that to an apparent rejection of all abstract math. — fishfry
I'm not enough of a physicist to comment. My point was only that you seemed to reject QM for some reason. I noted that you can't dismiss it so trivially, since QM has a theory -- admittedly fictional in some sense -- but that nevertheless corresponds with actual physical experiment to 13 decimal places. That's impressive, and one has to account for the way in which a fictional story about electrons can so accurately correspond to reality. Of course all science consists of historically contingent approximations. But lately some of the approximations are getting really good. Your dismissal seems excessive. — fishfry
FWIW I don't think anyone thinks the orbits are circular anymore. — fishfry
But you still have to account for the amazing agreement of theory with experiment. We might almost talk about the unreasonable effectiveness of physics in the physical sciences! — fishfry
I'm taking this from the end of your post and addressing it first to get it out of the way. As I mentioned, I didn't read any posts in this thread that didn't mention my handle. I only responded to one single sentence of yours to the effect that numbers are about quantity. I simply pointed out that there is another completely distinct use of numbers, namely order. Anything else going on in this thread I have no comment on. — fishfry
I think this helps to demonstrate that we cannot define numbers with counting. So, my original assumption that "2" implies a specified quantity of objects, must be false. But now we have the question of what does "2" mean? I think it is a sort of value, and by my statement above, a value we assign to empirical observations. However, if we can assign such a value to imaginary things in a similar way, we need a principle to establish equality, or compatibility, between observed things and imaginary things. This is required to use negative numbers. — Metaphysician Undercover
I may not be fully aware of the philosophical context of your use of "a priori." Do you mean mathematical abstraction? Because I am talking about, and you seem to be objecting to, the essentially abstract nature of math. The farmer has five cows but the mathematician only cares about the five. The referent of the quantity or order is unimportant. If you don't believe in abstraction at all (a theme of yours) then there's no hope. In elementary physics problems a vector has a length of 3 meters; but the exact same problem in calculus class presents the length as 3. There are no units in math other than with reference to the arbitrarily stipulated unit of 1. There aren't grams and meters and seconds. — fishfry
There's no time or space, just abstract numbers. I don't know how to say it better than that, and it's frustrating to me that you either pretend to not believe in mathematical abstraction, or really don't. — fishfry
You seem to want to deny the ideas themselves simply because they're abstract. That's the part of your viewpoint I don't understand. — fishfry
There is no need for time or space in math. I can't talk or argue or logic you out of your disbelief in human abstraction. — fishfry
You just phrase things like that to annoy me. How can you utterly deny human abstractions? Language is an abstraction. Law, property, traffic lights are abstractions. So is math. — fishfry
The notation is only suggestive of a deeper abstract truth, that of the idea of an endless progression of things, one after the next, with no end, such that each thing has an immediate successor. — fishfry
Now the set of natural numbers N={0,1,2,3,4,…}N={0,1,2,3,4,…} has no inherent order. — fishfry
You then stated that "we can only count representations of the imaginary things, which exist as symbols."
— Luke
That's a false quote. I said "we are not really counting the imaginary things, but symbols or representations of them". You said they only exist as symbols, not I. — Metaphysician Undercover
we can only count representations of the imaginary things, which exist as symbols. — Metaphysician Undercover
Luke, learn how to read! The representations, (which is what we count), exist as symbols. I did not say that the imaginary things exist as symbols. You've taken the sentence out of its context so that it appears possible that I might be saying what you claim to interpret. Though context clearly shows otherwise. This is exactly what I mean, you interpret, and represent what I say, in a totally incorrect (not what I intended), strawman way, solely for the purpose of knocking it down. Your MO, to ridicule, is itself ridiculous. — Metaphysician Undercover
imaginary things only exist as their symbols or representations, and if we are really counting those symbols or representations, then we are really counting the imaginary things. — Luke
imaginary things have no existence other than their symbols or representations — Luke
So do philosophers have to accept the actual infinite? — spirit-salamander
There don't need to be any real sheep in order to make the count. One could as easily count unicorns instead of sheep. Or Enterprise captains. Or any other fictional entities. — Luke
As I said, that's an order, one imagined thing after the other, it's not a quantity. — Metaphysician Undercover
:roll:Does that help? — sime
How is it that we can (really) order imaginary things, but we cannot (really) count imaginary things? — Luke
If "count" is defined as determining the quantity of, then it is an act of measuring. — Metaphysician Undercover
Or, just after Davis, Hancock, Carter and Williams laid down "Thisness" - an exceptionally gorgeous, introspective, haunting modern and abstract ballad - producer Teo Macero said to the group, "Your sandwiches are here." — TonesInDeepFreeze
You have nothing to order, and no order to offer. — Metaphysician Undercover
To say that the natural numbers have no inherent order, is to remove "order" as a defining feature of the natural numbers. — Metaphysician Undercover
Now we are left with quantity as the defining feature. Do you agree? — Metaphysician Undercover
I do not deny human abstractions, I just insist that they are fundamentally distinct, different from objects. — Metaphysician Undercover
I do not want to deny the ideas, I want to understand them. And understanding them is what requires spatial and temporal reference. The number 5 has no meaning, and cannot be understood without such reference. — Metaphysician Undercover
An abstraction must be intelligible or else it is meaningless, useless. If it can't be understood without spatial or temporal reference, then there clearly is a need for space and time in math, or else all mathematics would be simply unintelligible. — Metaphysician Undercover
And so the notion of elements isn't needed for the pure purpose of constructing abstract numbers. — sime
Category theory is a popular mathematical area. An offshoot of algebra, it can be used as an alternative to establish the foundations of math. It searches for so-called universal properties in various categories. Personally, I find it alien and entirely non-productive in the nitty gritty stuff I study in complex variables.
The Wikipedia page for Category theory gets 575 views/day, a respectable number. The page you linked gets 5 views/day and is classified as low priority (like my math page). So it may not help. But good try. — jgill
You are saying that counting is the same as measuring, but that can’t be right. Otherwise, what unit of measurement do we use to count? — Luke
The point is that by abstracting the concept of order from any particular meaning, we can better study order. — fishfry
The point of abstraction is to take away meaning such as first base, second base, so that we can study first and second abstracted from meaning. That doesn't make abstraction meaningless, it just means that we use abstraction to study concrete things by abstracting away the concreteness. — fishfry
Well, yes and no. Von Neumann's coding of the natural numbers has the feature that the cardinality of the number n is n. But there are other codings in which this isn't true, for example 0 = {}, 1 = {{}}, etc. So we can abstract away quantity too if we like. But that wasn't the point, Even if I grant you that cardinality provides a natural way of ordering the natural numbers, it's still not the only way. — fishfry
What do you call numbers, sets, topological spaces, and the like? — fishfry
But the 5 that mathematicians study is indeed an abstract object. It's not 5 oranges or 5 planets or 5 anything. It's just 5. That's mathematical abstraction. I guess I'm all out of explanations. — fishfry
There is no space or time in math. Why can't you accept abstraction? There's space and time in physics, an application of math. There's no space or time in math itself. Is this really a point I need to explain? — fishfry
The mathematician only cares about 5. — fishfry
It's certainly interesting that one can do set theory without elements. — fishfry
If I'm not mistaken, Von Neumann formalized without 'element' as primitive in 1925. — TonesInDeepFreeze
Counting is not "the same as measuring", it's a form of measuring. What is required for measuring is a standard, The standard for counting is "the unit", which is defined as an individual, a single, a particular. — Metaphysician Undercover
What unit of measurement is required for counting the natural numbers? Metres? Litres? Hours? Bananas? Obviously, no unit of measurement is required. You can count to ten without having to determine any unit of measurement. Therefore, counting is independent of measuring. Counting is not a "form of" measuring. — Luke
OK, so you define "order" as "having no meaning". That is your starting premise? What's the point? Any meaning you give to it will be logically invalid, as contradictory to that definition. There is nothing to study in a concept which has no meaning. — Metaphysician Undercover
Of course it makes it meaningless, you just said you take away meaning from it. If you take away all the meaning from "first" and "second", you just have symbols without meaning. — Metaphysician Undercover
If you leave some sort of meaning as a ground, a base, you have a temporal reference, first is before, (prior to) second. — Metaphysician Undercover
You are using "abstract" in a way opposite to convention. We do not "take away meaning" through abstraction, abstraction is how we construct meaning. — Metaphysician Undercover
There is a process called "abstraction", by which we remove accidental properties to give us essentials, what is necessary to the concept. We do not abstract away the meaning, we abstract what is judged as "necessary" from the concreteness, leaving behind what is unnecessary, "accidental". — Metaphysician Undercover
Sure, cardinality is not the only possible way of ordering numbers, but if the point is, as you described, to allow for any possible order, then we have to deny the necessity of all possible orders. — Metaphysician Undercover
That is to say that there is no specific order which is necessary. — Metaphysician Undercover
This removes "order" as a defining feature of numbers, — Metaphysician Undercover
because no order is necessary, so numbers do not inherently have order. — Metaphysician Undercover
Therefore order is not essential to the concept of numbers — Metaphysician Undercover
Then, we need something else to say what makes a number a number, or else we just have symbols without meaning. — Metaphysician Undercover
We could try saying that it is necessary that numbers have an order — Metaphysician Undercover
, but the specific order which they have is not necessary, like we might say a certain type of thing must have a colour, but it could be any colour. — Metaphysician Undercover
But this will prove to be a logical quagmire — Metaphysician Undercover
because it's really just a way of smuggling in a contradiction. — Metaphysician Undercover
It is impossible, by way of contradiction, that something must be a specific colour, and at the same time is possibly any colour. — Metaphysician Undercover
It is only possible that it is the colour that it is. — Metaphysician Undercover
Likewise, it is impossible that numbers must have a specific order, but could possibly be any order, because the order that they currently have, would restrict the possibility of another order. — Metaphysician Undercover
The point was, that if remove all order, to say that numbers are not necessarily in any order, then we must define the essence of numbers in something other than order. — Metaphysician Undercover
If this is cardinality, then cardinality is not an order. — Metaphysician Undercover
They are concepts, abstractions. — Metaphysician Undercover
I apprehend a difference between concepts and objects, because concepts are universals and objects are particulars. There is an incompatibility between the two, and to confuse them, or conflate them is known as a category mistake. — Metaphysician Undercover
It's an idea, and ideas are not objects. — Metaphysician Undercover
I have an idea to post this comment, and this idea exists as a goal. Goals are "objects", or objectives, in a completely different sense of the word. So if you want to say that numbers, as ideas are "objects", we'd have to look at this sense of the word, goals. — Metaphysician Undercover
But it doesn't make too much sense to say that they are objects in this sense, nor does it make any sense at all, to say that numbers, as ideas, are objects in the sense of particulars, because they are universals. — Metaphysician Undercover
Space and time are themselves abstractions, and these concepts very clearly enter into, and are fundamental to mathematics. — Metaphysician Undercover
Are a circle and a square not a spatial concept, which are mathematical? — Metaphysician Undercover
Is the order of first, second, third, fourth, not a temporal order whish is mathematical? — Metaphysician Undercover
If you seriously think that you can separate mathematical concepts from spatial and temporal concepts, then yes, this is something you really need to explain, — Metaphysician Undercover
because I've been trying to do it for many years and cannot figure out how it's possible. — Metaphysician Undercover
So please oblige me, and explain. — Metaphysician Undercover
The problem is that "5" means nothing without a spatial or temporal reference. — Metaphysician Undercover
If you think that the mathematician believes that "5" refers simply to the number 5, without any further reference to give the concept which you call the number 5 meaning, then you must believe that mathematicians think that the number 5 is a concept of nothing. — Metaphysician Undercover
If I'm not mistaken, Von Neumann formalized without 'element' as primitive in 1925.
— TonesInDeepFreeze
Would be most interested in a reference or more context. — fishfry
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