"1" does not refer to anything.
"1" has the superficial grammar of a noun, but this is misleading.
Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.
And of course this goes for other mathematical entities, too. They are things we do, not things we find.
Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.
And of course this goes for other mathematical entities, too. They are things we do, not things we find.
Comments (362)
Baloney. Are not ideas things we "find?" Whenever we discover a concept, is that not a "find?" Is a Hilbert space something we "do?" So there. :nerd:
No, they aren't.
Funny you brought up Hilbert, who was a proponent of mathematical structuralism.
'Hilbert said that in a proper axiomatization of geometry, “one must always be able to say, instead of ‘points, straight lines, and planes’, ‘tables, chairs, and beer mugs’”'
"Every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and ... the basic elements can be thought of in any way one likes."
https://www.iep.utm.edu/m-struct/
@Banno You editing a wiki article again?
But suppose there were a culture that said, "2 + 2 = 5," and their definitions of two, five, equality, and addition were the same as ours. If they all agree, they can't be wrong, can they?
Why would they do that?
Not wrong. Insane.
Rather, "Santa Clause" is to be understood through its role in separating people from their money near the winter solstice.
This goes for other entities as well. Superman, for instance, is a thing we do, not a thing we find.
I open a math book and find a new definition. Is that not a thing I find?
Metal detectors find buried coins. I suppose that is the naive notion you entertain. :roll:
Some one else put it there.
The split worm: 1=2.
Maybe. It comes from Wittgenstein. Do you think him naive?
This thread is a branch from @Sam26's Summary of the Tractatus.
It's as they say about guns: words and numbers don't refer, people do.
Not this time. Trying to read Wittgenstein’s Philosophy of Mathematics but I'm stuck on the way Rodych uses intension and extension.
So the extension of a set is the actual items in the set. The intension of the set is the rule by which one decides what is included - the property had my the members of the set. Rodych says that extensions must be finite.
Why?
One cannot physically list the integers. But in understanding the intension of "integer" we understand how to construct the extension... and in so doing it seems to me that we understand the extension to be infinite.
That is, whole there (presumably) is a StreetlightX, there is not a 1.
That's not what I'm after here. What you have constructed is a contradiction; they can't both claim to be adding in the way we do and that 2+2=5; we simply apply radical interpretation to work out which of the terms in "2+2=5" they are using differently, and change our interpretation to match theirs.
Tell me clearly what the word "find" means.
MW: "to discover by the intellect or the feelings" or "to come upon by searching or effort " or . . .
This thread is a good example of why philosophy appears sometimes to be "garbage in = garbage out"
When you start with a really shoddy definition things go downhill quickly. IMHO
:chin:
Quoting Banno
I made a small attempt to read him years ago but found little connection with the world of mathematics In which I lived.
Would you be able to take a look at 2.2 Wittgenstein’s Intermediate Finitism?
Can you follow the discussion of extension?
I suppose, but only because it's used that way right now.
I guess - to be less facetious - I don't want math to be anything all that special. It's a language too. The distinction in use between "StreetlightX" and "1" is intra-linguistic, and not between language and some other, special script.
But otherwise yes, "1" obviously doesn't refer to anything at all.
I've sympathy for that. I think it more like doing proper grammar than like metal detecting.
No, it doesn't. It doesn't refer to anything.
More or less. We invent, but we also discover. Creating vs discovering is a topic of interest occasionally for math people. Once we have invented we have set in play a process of unraveling or discovering what logically follows. Along the way we invent again, and follow paths stemming from those activities.
"it follows that 'the mathematical infinite' resides only in recursive rules"
As an analyst, I agree. I am not prone to use the infinity symbol like any other.
I'll read more later and report back. :cool:
Why finite sets? This seems an idiomatic use; and I;m not sure if it comes from Wittgenstein or Rodych
Well, yes - if one assumes that extensions are finite, then... extensions are indeed finite.
And my apologies for baiting you with the OP. I needed a mathematician who might disagree with a constructivist approach to mathematics; the ruse worked.
Why not?
Quoting Banno
Same as my question to Baden. Why do we assume that radical translation must yield the same thing we have?
And to both of you: I'm not just being difficult. It's all quite relevant, I think.
Charity.
My dog does one’s and two’s. The one’s I can generally ignore, but the two’s I have to clean up.
I read the article on Witts philosophy of mathematics some time ago, Wittgenstein rejects the set of real numbers or any sort of infinite mathematical extensions.
Extension (concatenation of symbols) will always be finite. Mathematics is all the combined knowledge of intentions and extensions and nothing beyond that. We can only understand infinity as an intention. There is an infinite possibility of natural numbers, not a set of infinite natural numbers according to Witt.
I can tell where he is wrong though, and it's here
Quoting Banno
I think you've made a use-mention mistake there.
1. 1 is greater than 0
2. There is a '1' in the previous sentence
The '1' in sentence 2 refers to the '1' in sentence 1, even if the '1' in sentence 1 doesn't refer to anything.
Another mistake you may have made is conflating numerals and numbers. Numerals refer to numbers, even if numbers don't refer to anything.
So when you say "'1' does not refer to anything" are you referring to the numeral or the number? If the former then I think you're wrong (the numeral refers to the number). If the latter then I think you're arguing against a position few would take (numbers, like tables and cars, are commonly thought of as referents, not referrers).
If "a" refers to "b" , doesn't this imply "b" refers also to "a".
Even if you do not explicitly state what the number "1" in the sentence 1 refers to. After you have established a connection to sentence 2, there is an implicit relation.
1. I am a man.
2. The last word in the first sentence has three letters.
"The last word" in sentence 2 refers to "man" in sentence 1 but "man" in sentence 1 doesn't refer to "The last word" in sentence 2.
What, to explain,
Quoting Banno
? :chin:
If counting is something we learn, then counting is something we find.
If we aren't pointing at individuals when counting, then what are we counting, numbers or individuals?
"1" refers to the individual counted first among the counted.
"One" is a noun (or, more precisely, a pronoun) in the sentence, "I'll have one," where "one" is whatever you were referring to.
In the sentence, "I have one dog," "one" is an adjective and there is no referent to the adjective other than it being a descriptor of the dog. You reach a similar result with other adjectives, as in, "I have a happy dog." Happy is not a thing.
So, give me some examples of where "one" is superficially a noun where you're just not identifying "one" being used in the adjective case.
... in whatever the discourse. Charity and world-domineering ambition alike require translation between discourses, and agreement re ontological commitment (re, e.g., what "1" refers to). Pedagogy and practicalities require, instead, toleration of alternative systems.
What is the meaning of 1 if not pointing to X?
1 counted, is 1 count where X = count. 1 without X is 0.
"We can ask whether numbers are essentially concerned with concepts. I believe this amounts to asking whether it makes sense to ascribe a number to objects that haven't been brought under a concept. For instance, does it mean anything to say 'a and b and c are three objects'? I think obviously not. Admittedly we have a feeling: Why talk about concepts; the number, of course, depends only on the extension of the concept, and once that has been determined, the concept may drop out of the picture. The concept is only a method for determining an extension, but the extension is autonomous and, in its essence, independent of the concept; for it's quite immaterial which concept we have used to determine the extension. That is the argument for the extensional viewpoint (p. 123)."
We have a concept (a mathematical concept), and we use the concept to refer to things, but the things do not reflect the concept, i.e., it is not as though the concepts are intrinsic to the things. We group things together under the rubric of the concept, and we extend this concept to group other things under the same umbrella. "The extension is autonomous." The extension reflects a certain state-of-affairs that is brought under the mathematical concept.
Are letters objects? Are ink scribbles on paper objects? Are symbols objects?
Letters and numbers are each individual objects that can be counted. How many numbers are on this screen? How many letters? What are you counting when answering this question - objects or what?
You tell me, do you or we refer to marks on a piece of paper as objects? I think not. Some might say that they refer to objects.
Is that settled science?
This depends on how you understand "1". You can understand it as playing a role in counting, as you describe, in which case we can assign some sort of priority to it. But many modern mathematical axioms remove this priority, denying that priority, assuming that mathematics is something other than a tool for counting. Measurements are not all instances of counting because we employ negatives etc..
But you can also understand "1" as a fundamental unity, and this gives it a logical priority, as the designation of an object, a unity. This is required for all logical processes which proceed from the assumption of objects.
The two forms of priority are not completely incompatible though.
Quoting Banno
The concept of "set" requires the assumption of objects. So set theory utilizes "1" as a unity, an object. This is an issue with set theory which I discussed with someone else in another thread recently. To assume that a set has extension is to assume Platonism, because it necessitates that a number is an object, being derived from that assumption. As I argued in that thread, "infinite extension", which is what conventional set theory allows, is incoherent, based in contradiction. An object, as a unity, being unbounded, is fundamentally contradictory.
What I found in that other thread, a conclusion you may or may not be interested in, is that there is an issue with the usage of the law of identity, in formal logic. The law of identity, as designed, is intended to assign uniqueness to an object. The law of identity as employed in formal logic designates a form of equality. Equality and identity are distinct ideas. Identical might be a specific way of being equal, but being equal does not necessitate being identical. In formal logic, the latter is assumed to be the case, that being equal is the same as being identical. So the law of identity, as originally formulated, is violated by formal logic, which employs a distinct interpretation (misinterpretation) of it.
What would your take on a formal semantics approach to 1's referent be? Like, taking it to be by definition the successor of 0, or the equivalence class under bijections of { { } }.
What are marks on a piece of paper, if not marks of ink, or lead? Does ink cease to be an object when it gets transferred from the pen to the paper?
Symbols are objects used to refer to other objects. A stop sign is a sheet of octagonal shaped metal with red and white paint, that refers to the act of stopping one's vehicle.
Different signifiers, same signified?
If 1 is to be understood through it’s role in the process of counting, wouldn’t it refer to a specific point in that particular sequence?
1 is simply an abstraction most beautifully captured with set theory as the property shared by sets of type: {a}, {#}, {£}, etc. (sets with only one element).
Wherever we deem some such, like we talk about and point at things every day, we say there's (a quantity) of 1 of that, regardless of whatever exactly it may be.
Could include hypotheticals and whatnot, too.
It'll take some conceptualization to get 2 (of those), but we may instead have 1 of this and 1 of that, thus having 2 of this or that together.
So, in the abstract, 1 (just 1) would denote 1 of anything, without referring to anything in particular, but still exemplifiable.
Seems to be how we typically use 1 anyway, no?
Thank you. I'm picturing this as direction of fit, as in Anscombe. That the number of things in the box is 3 is something we do; the direction of fit is from us to the world.
Add to that, that concepts are not things so much as a way of behaving; that is, concepts are best not considered as things in people's minds, but as ways of talking and acting. (compare street's recent thread on emotions as concepts)
Then we have a way of talking that goes "one, two, three" while pointing to each thing in turn. And we can use this way of talking to talk about lot of different things. And then we can talk about this way of talking when we find ourselves adding, then multiplying, then differentiating...
SO even though numbers are not things, we develop mathematics by treating them as if they are. And in the end they become thigns just by our having treated them as such.
So, the extension of "Sam26" is Sam26. The extension of "red" is each and every red thing. But the extension of "1"? Well it's literally every individual. And as such it seems to me, at least in my present mood, that the extension drops out of the game, and what we have is the intension, the rule, concept or game we play in counting.
And the consequence of that is that talk of extension in mathematics becomes fraught with ambiguity. Hence, Wittgenstein's argument that mathematical extensions must be finite, and hence his adoption of finitism, seems misguided.
I hope I explained - in outline at least - that abstract entities come form our talking about what we do; so in talking about counting, we pretend that integers are real things, and this leads us on to more complex ways of talking about integers, and so a sort of recursion allows us to build mathematics up from... nothing.
What?
I presume you do not mean that abstract entities are generated from our talking about what we do, since that would make them things anyway. So what do you mean by "come from"?
But one of the things we do, is to talk. And we can talk about our talk.
Every few years I set up a forum game in which players take it in turns to add a new rule. Offten the consequences are mediocre. Sometimes they are extraordinary.
I don’t get it. Maybe I’m counting wrong, because I’ve never used “1” to refer to the third count in a counting sequence.
Yeah, I noticed that.
Re Wittgenstein's finitism, for me it always just fell out from his view that mathematics is nothing over and above a human activity, and since we are finite, nothing we can construct is going to be infinite. Not sure whether that's a good argument or not, but it seems to be the bare bones of it. The idea that sets have only finite extensions is in any case not generally accepted, ZF set theory even has an axiom that includes infinte sets from the outset. Of course, W's reading of this would presumably be that the axioms just give recursive rules we can use to continually come up with new, disinct members to add to a set, but we always have to stop doing that at some point and just say "and so on" or, the more mathematically acceptable, "...". As you said, sure extensions are finite if you define extensions as finite, and Wittgenstein defined them as finite, but others did not. He thought they were making a mistake. They did not.
I'm flummoxed that Wittgenstein's argument might be so artless. And so, I'm asking for something more.
So we have him baulking at the the diagonal and rejecting incompleteness as a result. Yet I am in agreement with his constructivist views, as set out above. I was bothered that the one might necessitate the other; however it seems now that they are unrelated. At least, no one here seems to have shown such a relation - but then so many of the replies appear not to have been on topic.
So I might be wrong.
@jgill? What sort of thing are numbers?
I get the sense from the SEP article that Wittgenstein will allow what modern philosophers of mathematics call potential infinites. It looks like Witty would be okay with, e.g. saying that the successor function can be applies an infinite number of times.
I'm not sure how problematic this really is. "There is no set of all the real numbers" is only true from this perspective if we regard the set as an extension. What's stopping us from just paraphrasing it as shorthand for what can be done with an intention and a finite extension(s)?
EDIT: nevermind, I derped. Obviously you can't do that with the reals 'cause they're uncountable. So you have to junk the reals.
As I've suggested, I don't see how constructivism is committed to finitism
Nor do I see that constructivism is committed to rejecting the Law of Excluded Middle. Rather, a constructivist approach would say that including the law leads us this way, excluding it leads us that way; and which way you choose depends on what you are planning to do.
Another way to put it is, contrary to Brouwer, mathematics just is the language of mathematics.
Nor is mathematics just a creation of the mind; that's too solipsistic. Mathematics is a collaborative enterprise, not something in individual minds.
But you said that those two things are unrelated. How come? How do you save the set of reals if it can't be constructed recursively?
There's an anecdote attributed to Wittgenstein showing how an infinite past seems uniquely counterintuitive.
Wittgenstein overhears someone saying "5, 1, 4, 1, 3. Done."
He asks what that was about, and they respond that they just finished reciting ? backward.
"But, how old are you?"
"Infinitely old. I never started, but have been at it forever and finally finished."
Not logically impossible or inconsistent, notes both James Harrington and Craig Skinner, but a strong intuitive argument nonetheless.
The moment they were done reciting seems random, there seem to be no sufficient reason their recitation was done at one time and not another, any other. And likewise for any of the other digits.
So, with our expectation violated, we tend to reject the thought experiment, and out goes an infinite past.
Anyway, looks like Wittgenstein doesn't accept ? as such.
Not sure I buy the rationale here, but intuitionist physics is a worthwhile pursuit I think:
Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.
[i]Natalie Wolchover
Quanta Magazine
Apr 2020[/i]
I take that as an invitation. Equality always requires a qualification, the same quantity, the same quality, the same size, shape, degree, etc.. Without that qualification "equal" is meaningless. Identity indicates "the same" absolutely, without qualification. The former, equality, Is an identification by means of reference to properties. The latter, identity, is an identification of the object itself, regardless of properties. Can you apprehend the difference between pointing to an object, thereby identifying that object without reference to any properties (identity), and stating such and such properties, and finding whatever object, or objects, which are indicated by (equal to) that description.
Notice Tractatus 4.12721, a multiplicity of objects fall under a formal concept, so a formal concept itself cannot be an object. A number "1", "2", as a formal concept cannot be an object. The law of identity indicates an object, it cannot indicate a multiplicity of objects which are equal by the terms of the concept, which is what is signified by a formal concept.
"1" refers to an abstraction , as do all natural numbers.
Consider the abstraction, "3" . Three-ness is a property that is held by some states of affairs: those consisting of 3 objects.
"+1" is a successor relation that holds between two consecutive natural numbers. In the real world, this maps to a relation between states of affairs. For example it's a relation between a collection of 3 apples and a collection of 4 apples.
This may work in some weird way, but this would also bastardize the language. The concepts "point, straight lines, planes" have at least some semblance to human envisioning what these things mean in geometric concepts. While "tables, chair and beer mugs" would also work if used consistently, there are already assigned meanings for these words that are vividly different from the assigned approximate meanings of point lines and planes in our language.
In other words, a person could rewrite entire books of science, philosophy and literature, by assigning to each word's meaning a totally different existing word, which would lose its original meaning. This is good exercise is logic and in theoretical thinking of the use of language, but would amount to nothing more. Therefore it is not done. Notice, that no textbook of geometry uses "tables, chair an beer mugs" for "points, straight lines and planes." There is no other reason for the lack of wicked bastardization, but the fact that some words are more conducive to conjure up a meaning for a newly introduced concept.
One is the sound of a single finger snapping. :cool:
Like staring at the sun, looking too hard into the foundations of mathematics can damage the mind's eye.
In a running race (like Marathon races) people are assigned each a different number. For identification purposes.
Would you call those numbers (one included, and other mathematical entities included, such as "2", "4", etc.) things we do, not hings we find?
The runners could be called "A" "B" "C" ... "AAZAET", etc. or they could be identified with a scale of colours.
This opening post has pretended to define the true nature of "1", but alas with impoverished thinking. Language uses its components in many ways, and to try to restrict a multiply-used component to fewer uses than the language already employs for that component, is a proposition that is obviously wrong.
Let me explain. The Opening Post appeals to the masses to use the word only in the meaning that the writer of the OP allows. But the word has long ago grown beyond that meaning only. The OP ignores other valid meanings to prove its wrong point, and declares the other valid meanings wrong. This amounts to nothing less than trying to redefine the language.
That's why they finished when they did. That point in time when they actually finished was just as valid as any other point in time to finish, since any other point in time would have been equally as valid a finishing point as the actual one.
The reason that is sufficient to explain why they ended when they did, is that 1. They could have ended any time, reasonably, and 2. the time they ended at was in the set of "any time", and 3. unsaid, but assumed, and fulfilled the requisite, that there is only one time that the recitation ends. It can't end, for instance, two different times. Or 345 different times. It can only end one time.
I do not think you are likely to get it. As far as I recall Wittgenstein himself did not in the end think very highly of his lectures on the foundations of mathematics.
What is a pointer?
It's something that points a mind to an object/subject.
Like a lazer pen - not to say we have cycloptic vision.
I never actually agreed with the number's significant role.
1 points to 2-9 in base 4 and 0 can be understood relative to 1, base 4 is a category of 1, and so is base 8. 1, has it's use.
In this case it's not a pointer but a medium of communicate object/subject relativity. That is 1X, interpreted impurely.
It's just not so significant...
Sure, maths as fiction with a super-coherent plot.
And with illustrations, too. Kind of, Alice in Wonderland.
Yet Sam26 was able to make perfect sense of what I said, and everyone else I have had a conversation with on these forums, was able to make perfect sense of what I said. You are the only one that has a problem making sense of what I say.
Quoting Banno
But when others can't make sense of what you said then that's their problem. :roll: I said the same thing as NOS4A2, yet you understood them. These are the symptoms of a delusional disorder.
Quoting Banno
...the same number of what? In starting anywhere, you'd change the context of your counting, and would be counting something different, so how would you get the same number?
One could argue, probably successfully, that Wittgenstein was not a finitist, i.e., he never held to the idea that the finite character of language meant that there weren't infinite processes or methods. He was mainly interested (at least it can be argued) in the problem of the grammar of the infinite method or procedure. In other words, how is it that finite signs, as expressed by finite beings, have a sense of infinity. This has more to do with Wittgenstein's later philosophy, i.e., what it means to master a technique or practice.
Forum members who've spent any amount of time in dialogue with Banno know he's all about force and politics. (For all his emphasis on a foundational charity.)
True. Wouldn't that be appropriate for a behaviorist?
People esteem 1 too much when it's clear that we have done wrong by it.
It would be better if 1 didn't exist so referring to it using 'it' would be false.
This is why I said 1 is a pointer, to imply axiom is use.
That, and playing language games.
If using numbers and words doesn't entail using finite objects to refer to other finite objects, then Banno isn't talking about or counting a number of anything. He would just be making ink marks on paper or making sounds with his mouth when "counting".
More language games.
How can you even say that one follows from the other - that one gets a sense of infinity from finite signs expressed by finite beings?
Where do you think our sense of infinity comes from? It comes from us, i.e., finite beings, we create the concepts using finite signs. We extrapolate based on the continuation of 1,2,3.. that it goes on ad infinitum. There's no mystery here.
There is a means of grasping a rule that is not an interpretation, but is exhibited in following and going against the rule in actual cases, says (paraphrased) Witty.
Did he have anything to say about the Halting Problem? I have a sudden, strong hunch that it's related to this. Maybe I'm just seeing things, but grasping that a Turing machine goes on forever without doing any calculations seems to be a case of grasping a rule in Wittgenstein's sense.
I don't know Pneumenon.
I believe I've heard Banno say language is non-referential. Words don't refer to anything. So why would "1" be the exception?
Yes. However see 6.211, from the Tractatus, in which he talks of mathematical propositions being nothing unless used. That's gotta be a harbinger of things to come.
So he certainly would not have gone along with the finitism of @Metaphysician Undercover who rejects instantaneous velocity.
But,
and
So it's being argued that "1" has an extension, while "root 2" does not - that "1" pints to 1, while "root 2" points to a recursive rule for generating an infinite decimal. However I'm thinking, as posited in the OP, that neither has an extension.
I may have been a bit rough on Harry; in my defence, on the occasions in which I have engaged with him, not much happened.
If it is a pointer, it can be used to point to anything.
Which seems odd.
Quoting Harry Hindu
Sometimes we do talk about infinity. When we do this, we are using finite objects - ink marks and sounds.
So...?
I'm not following this at all. Are you claiming that we do not talk about infinity? OR that such talk is no more than sounds?
, ? What's this about?
See 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability
SO a Turing Machine could be set up to calculate 1+1, and would halt - hence 1+1 has an extension; but if set up to find root 2, it would not, and hence root 2 has no extension... or something like that.
Banno thinks that "Banno" can be used to talk about Banno.
Nothing controversial about that.
Language can be about stuff. It's just that it can do other things as well. This in contrast with what might be @Harry Hindu's view - it's hard to tell - that language is only about...
I'm sure Harry Hindu agrees language can both be about stuff and do other things as well.
But I remember a thing about the non-referentiality of the T-sentence. I possibly assumed if the T-sentence is non-referential all language is.
Then there was the thing about the uselessness of a non-referential T-sentence.
Does a word refer?
Think on that question. I've already said that "Banno" (a word) can be used to refer to Banno.
SO what is it you are asking?
So "Banno" refers and "1" doesn't.
Maybe so. Maybe not.
Maybe so and maybe not (Maybe so and maybe not)"
Stash, Phish
https://www.youtube.com/watch?v=1BJlSU308Wc
The reason for rejecting "instantaneous velocity" has nothing to do with mathematics, the notion is self-contradictory. Velocity is distance covered in a period of time. There is no period of time at an instant. There is no distance covered at an instant. There is no velocity at an instant. There is no "instantaneous velocity". No matter what sophistry the mathemagician might apply, the smoke and mirrors cannot hide the contradiction from a trained philosopher.
So the existence of potential infinites is secured by our ability to grasp a rule, and that rule becomes an intension in the sense used in the SEP article. If the rule allows to construct a finite extension, then we can get extensions from it, too.
So the extension of the set of integers is always finite, although it can be continued arbitrarily. And now I'm being assaulted by that giddiness of logical legerdemain that Witty talks about...
Maybe this is swinging too hard, but: the motivation for this eludes me. Abstracta are spooky, but so are ineffable rules grasped without interpretation. Why does Wittgenstein like this spook more than the Platonic spook?
This is the bit that I've been unable to find clearly articulated. But it seems to be what is being argued.
Quoting Pneumenon
It seesm to be...Quoting Pneumenon
Yep, so you have said.
And yet, we can Calculate Instantaneous Velocity
So we conclude that either physics is wrong, or Meta is wrong.
Okay, let's try an example: the successor axiom in Peano arithmetic says that if a is a number, then so is its successor. And the induction axiom says that if s contains 0, and also every successor of every one of its elements, then s contains all the numbers.
So does Witty's constructivism make the induction axiom nonsense, or does it mean we have to construct the induction axiom from an intension and the number 0? The successor axiom, presumably, is or contains an intension.
For my own part, I'm thinking that the extension/intension juxtaposition in this context is ill-defined and confusing... or it might be just me. Anyway, hence the OP; that "1" does not have an extension; or rather that talk of extension/intension is misplaced in mathematics.
That's because you confuse stopping a particle at a specific time and observing a particle at that time. Don't forget momentum. :roll:
Just to be clear, are you both dropping (or taking as read) an "infinite"?
* typos
From your referred article:
"However, this technically only gives the object's average velocity over its path."
As I said, smoke and mirrors. Neither meta nor physics is wrong, Banno's misled by deceptive word use.
Quoting jgill
One cannot observe an object at an instant in time. An observation occurs over a period of time. I don't see how "momentum" is relevant. Remember the uncertainty principle?
That sentence refers to the v = s/t formula.
If it takes me 10 seconds to move 10 metres then my average velocity is 1m/s. But it may be that my velocity was less than 1m/s for the first 5 seconds and greater than 1m/s for the last 5 seconds (because of acceleration).
Come on, Meta.
Right, so what does "instantaneous velocity" mean?
The website says this:
" This is called instantaneous velocity and it is defined by the equation v = (ds)/(dt), or, in other words, the derivative of the object's average velocity equation."
The "derivative" is an approximation which creates the illusion of compatibility between a period of time and a point in time. This is evident from the fact that it is a differentiation. So the "instantaneous velocity", is derived from a period of time, and presented as a point in time. If someone believes that it is a true representation of a point in time, that person has been deceived
You seem to be one of those someones, who has been deceived by the smoke and mirrors.
Imagine you're driving and you watch the speedometer go up and down as the car speeds and slows. The instantaneous velocity is whatever it shows at a particular moment in time, e.g. if you took a picture.
Do you have any expertise in maths or physics? I don't, but I'm pretty sure derivatives and instantaneous velocity aren't just "approximations" and "illusions".
No matter how you look at it, "instantaneous velocity" is an average, and does not represent a moment or instant in time, in any sense of "true" representation, unless "moment" or "instant" is defined as a period of time.
Quoting Michael
Even taking a picture occurs over a period of time. A camera is not capable of stopping the clock at a point in time, to show how things would appear at that point.
Quoting Michael
Do you know what a differentiation is? It requires two distinct descriptions of the same changing thing. Therefore the possibility of a single point in time is excluded.
Sorry, Meta, that's just wrong.
Nice argument Banno. Unsupported assertions are a sign of ignorance.
Quoting Banno
1 has it's beauty, it's not this represention we have of it, detrimental to all and stupid.
A real complexity of the mind is shown here. If we can make ourselves stupid by believing in 1, what is the sense of good?
How 1 math is used results in incorrect measure of velocity, but it is sufficent complementory when such building a portofolio.
That's what I understood from this thread.
Causation (i.e, first cause, god, no beginning and no end). The fact that you "start anywhere/somewhere" should be an indicator that you're not dealing with infinity when counting.
Ever looked into a mirror that is across the room from another mirror, like in a dressing room? What about a circle? The Greeks were the first to mention infinity as a boundless system. Aristotle argued that there was no actual infinity, only potential infinity, which I interpret as imagined infinity.
So, what does counting, and numbers - of which only a finite number have ever been written or conceived, or used, have to do with a conceptual paradox we call, "infinity"?
What you mean, is that we calculate something called "instantaneous velocity", which employs a faulty representation of "instantaneous" in relation to the philosophy of rigorous definition, and you falsely assume that this is a true representation.
Ever wonder why physicists cannot determine the position and momentum of a particle at the same time (uncertainty principle)? Perhaps you ought to consider that it has something to do with the principle I'm arguing, the mathemagician's representation of velocity at a point in time is not at true representation.
Quoting Banno
Quoting Banno
I would argue that language being about stuff is language's primary, if not it's only, function - to inform, to communicate. I would also say that our concepts are what language is about and our concepts are either about the world or aren't (objective or subjective), and that sometimes it is difficult to impossible to distinguish between the two.
So, we can talk about infinity like we can talk about God or talk about Queen Elizabeth. When it comes to "infinity" we're not sure whether it's only a concept or a fundamental feature of reality (or potential vs. actual infinity as Aristotle put it) just as we're not exactly sure if "Big Bang" refers to any real event that occurred before the existence of beings to conceive it and use scribbles and sounds to refer to it.
Now, if you could offer some examples of "language use" where words and numbers are not used to inform or communicate, and does not equate to just making scribbles and noises, then I would be interested in talking about those cases.
Only odd if you were the only being in the universe. You wouldn't be using pointers because there would be no need to point to things if you were the only being in the universe.
The fact that we live with others that have views of the world like we do, but might have false beliefs, or missing views that you have, gives us reason to point to things to inform, to communicate. But we have to agree on the pointers to use and what they point to. Different cultures use different pointers to point to the same thing, which is what we are translating when translating languages - what the pointers are pointing to.
When I'm at home alone, playing a game and losing, I often shout out "for fuck's sake". I'm not informing or communicating with anyone. It's an expression of frustration, much like laughing is an expression of happiness and crying is an expression of sadness. I wouldn't say that any of these expressions point to or are about anything (in the sense of reference). They may indicate something, but that's not quite the same thing – talking fast indicates that I'm in a hurry, but that doesn't mean that my words refer to the fact that I'm in a hurry.
So if numbers are an aspect if counting, and one cant count to infinity, then finitism.
But if we reflect that what a word refers to or is pointed at is never a matter of fact anyway, but is one rather of interpretation, theoretical parsimony then strongly argues against the easy option of distinguishing as many varieties of meaning as we might have different words for. Obviously no two of these kinds will ever be quite the same thing.
The argument isn't just about theoretical desiderata and separate from the subject-matter: the behavioural interactions we are discussing depend on agents' anticipations of each others' interpretations, so we are theorising about theorising (about...).
And so I applaud @Harry Hindu's objection here to the habitual distinction of expression and exhortation from description. My attempt here.
To expand a little: since no bolt of energy (nor any more subtle physics) connects uttered word to object, we (interlocutor or foreign linguist or even utterer) are perhaps entitled and perhaps required to interpret the utterance as pointing, in various degrees of plausibility, not only a presently uttered token but also the "word as a whole" at (not only a present object but) some kind as a whole, and then by implication as also pointing not-presently-uttered but semantically related words at related objects and kinds. In other words, any speech act offers a potential adjustment (or entrenchment) of the language in use, so that the extensions of related words are shifted in related ways.
Hence utterances that vent frustration can also offer (directly or indirectly) potential adjustments to the extensions of words ("patient", "skilful" etc.) that might or might not point at Michael.
And hence also Harry's and my other examples as linked above.
Number 1 as a symbol on the screen is free from all meaning. We ourselves give meaning to it and see it as a representative of something that we might imagine 1 is pointing towards. Why would number one have any inherent value? Also, we can't exactly count objects in the universe. Is atom one or made up of lots of smaller parts? Where do we put a boundary between one object and another object? We can't use numbers to simply quantify some objects without having complete knowledge of what that object is.
That's a bit too fast, but in being wrong, might be the gist of what is going on. It's worth talking to a child about infinity to see the change in thinking as they realise that for any number they construct, someone can make a bigger one; they say "a squillion billion", you say "a squillion billion plus one". Then the confusion when they begin to realise that "infinity plus one" is still infinity. The game changes before them.
So they come to realise that for every integer there is a bigger integer, and despite that we can talk about all the integers. It's the sort of recursion that recurs in maths.
Picture a child saying "but you can't talk about all the integers...Quoting Metaphysician Undercover(misquoted)
...and claiming finitism.
So the rule is that for every number, one can add one. The rule only generates one new number. One has to see the rule in a different way in order to understand infinity: imagine a number bigger than any number the rule could generate...
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Anther way to approach it that the rule "For every number, you can add one. to make a bigger number" is not generating all the numbers, but only the integers. We can find infinity by calculating 1 divided by 3, as a decimal; or by asking what number times itself makes 2.
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SO we learn how to count, and then we learn how to do other things with counting.
But the salient point is that some utterances do other stuff as well.
(*Attributing propositional content to "Hello" seems excessive...)
Well, that's not right, is it? We count things all the time.
A finitist has no problem with speech about infinity. The spaceship travels on forever. The reading on the odometer is always finite. That's all.
Where's @Nagase ?
It doesn't. It just doesn't allow talking about the infinite as if it's finite (IOW set theory).
"1" represents an idea of quantity. Does it follow that "1" refers to an idea of quantity? I would say the answer to that depends on what you mean by "refer".
Quoting ZzzoneiroCosm
Being a pedantic arsehole, I am driven to correct you on this: "1" refers to 1.
All you've done is offered two distinct definitions of "number". Under the first definition, we get a bigger number by adding a number. Following that rule, there is no way to get a number bigger than what is given by that procedure. Under the second definition, we must assume that there are other numbers, not derived in the first way. The bigger number in the second definition will never make it into the first set of numbers, so the two are in that sense incompatible. It's not a huge problem, just like different possible worlds, so long as we recognize the points of incompatibility. If the incompatibility goes undetected there may be a problem because each is named "number".
Here's the issue though. We can count anything we would ever need to count using the first rule, so what's the point of the second? It doesn't help us to count anything, all it says is that no matter how high we count we can still count higher. But we already knew that, because we know that we can keep adding a number. So it doesn't allow us to do anything more than we can do with the first rule, nor does it tell us anything we didn't already know from the first. It is completely useless, and on top of that it gives us a new kind of "number" which is incompatible with the rules in the system of counting. Looks like an axiom designed for equivocation to me.
Quoting Banno
You're looking at this in completely the wrong way. A whole number is undivided. The integers are a special formulation of whole numbers, allowing for the inclusion of zero and negatives. Now you want to divide these whole numbers into parts. These are fractions. So why not call them "fractions", because that's what they are? Instead, you want to call them "numbers". Same problem as above, we now have some sort of numbers which are incompatible with the other "numbers". Why do that? You're just creating confusion and a recipe for equivocation again. If the "numbers" are the counting numbers, and we can (in theory) divide these numbers into parts, then why call the parts "numbers" as well?
Do you recognize that "one" is a fundamental unity? If you divide one in half, this does not give you two, it gives you two halves. Why would you want to represent a half as a number, when it's clearly not a number, it's a half? Some flamboyant mathemagician artist comes along and says let's make an axiom whereby a half, along with all other fractions become numbers, wouldn't that be cool. No it wouldn't be cool because there's a big problem, some fractions cannot be represented as numbers. Instead of recognizing, well that was a mistake, let's leave a distinction between numbers and fractions, the mathemagicians just try to cover up the mistake with more and more complex axioms.
Quoting Banno
This not quite right. You should say that we learn how to count, then we learn how to do other things with numbers. The other things are not counting. We could call the other things "art", but a lot of it is more like a magic show, illusions, smoke and mirrors, deception.
Yeah, it's kind of fully sick Zen.
Well, with a bit of work it allows us to find an instantaneous velocity... among other things.
The reification consists in the idea that there are an infinite set of numbers. The ordinary procedure of counting has no logical limit to it; so, it can go on forever and hence comes the illusion of infinite series. Counting can go on forever, but there is never a point where infinity is attained, or even more closely approached.
Smoke and mirrors.
You keep saying that, as if it were an argument.
But we have:
Quoting Metaphysician Undercover
And yet, we can Calculate Instantaneous Velocity
So we conclude that either physics is wrong, or Meta is wrong.
My thought was that "1" might refer to "1". A circularity. Not that "1" might refer to 1.
If "1" refers to an idea, then it is an idea shared. Else your idea of 1 would not be the same as mine.
So what sort of thing is that?
I made the argument, and addressed your reference.. You rejected my argument with nothing more than "you're wrong". Sorry but it's you who has presented no argument.
I explained already, the uncertainty principle demonstrates that physicists are not really calculating instantaneous velocity. Physics is wrong, they are not calculating instantaneous velocity. They might call it that, but it's clearly not what it is. Otherwise there'd be no uncertainty in the question of the momentum of a particle when it is at a specific place at a specific point in time.
One thing at a time?
Quoting bongo fury
To which,
Quoting Banno
Ought that have clarified for the competent reader that @Pneumenon meant "then we can get infinite extensions from it, too"?
Just hoping not to misunderstand either one of you.
We were trying to decide if the version of Wittgenstein in the article thought there couldn't be infinite extensions... I think...
Which is why I said Quoting Banno
No! Only whether the word-string "then we can get extensions for it" was a misprint of "then we can get infinite extensions for it"?
A different reading of it (as not a misprint) seemed plausible, so I thought I should check.
And ...?
You're the one who changed the topic. Instead of wanting to discuss the issue, what it is that is represented by the formula they call "instantaneous velocity", you changed the subject to a question of who's wrong, physics or meta.
Your refusal to address the issue is getting rather boring. Instantaneous velocity is an average, we went through this yesterday. There is no such thing as a determination of velocity at a point in time. That's obvious, nothing moves when no time passes, so to determine any velocity requires a period of time. If this does not make sense to you, and you won't take it from me, do some reading as to what "instantaneous velocity" really is, it's an average.
https://openstax.org/books/university-physics-volume-1/pages/3-2-instantaneous-velocity-and-speed
Notice the decisive phrase, the time "between the two points approaches zero". If it was truly an instant, there would be no time, the value for t would be zero, and the equation would be useless.
A sort of thing that is.
The page you referenced quite explicitly sets out the difference between average velocity and instantaneous velocity.
Fuck, there's even a diagram.
Abstract object.
The fact that someone can add one to some number in no way implies some notion of infinity. If anything, adding one to some number just produces a finite sum, not an infinite sum, hence my mention of Aristotle's actual vs potential infinity. Potential infinity is an idea that can never be realized as actual infinity.
"Infinity plus one" is incoherent. Infinity would already include all the ones, twos, threes, - everything. So by adding one to infinity implies that infinity wasn't infinite in the first place.
That's finitism I believe.
Right, so in your example, you'd be just making noises with your mouth.
Like I said, any instance where you aren't using words to refer to something, or to inform you of something, then you're just making scribbles or noises, but then making scribbles and noises are themselves about, or can inform someone of something.
If I can determine that you are in a hurry by the way you speak, and not what you said, then if you had said, "I'm in a hurry" wouldn't that have been redundant being that you communicated you being in a hurry by the way you spoke? We often communicate without knowing it using body language. Talking is just another form of body language, of what certain bodily behaviors can be about. With words, we've simply added another layer of aboutness. Not only can I determine that you are speaking, and that you understand English, but your words are themselves about other things - another layer of meaning. I could tune out what you are saying and focus on your lingo and use of the language if that were my goal. Information is everywhere and is causal. The information I can ascertain from some effect (like your typed words) is the relationship that effect has with all the causes that lead up to it (like what you know and your understanding of the language you are using), it just depends on which set of causes we are focusing our attention on at the moment.
Only that our thoughts are finite. We don't know if the universe is.
But then this brings up how our thoughts relate to the world. In thinking about infinity, do we really need to have infinite thoughts? In thinking about some thing, whether it be an apple or infinity, do our thoughts ever exhaust what it is that our thoughts are about? Is it preposterous to assert that in thinking about an apple, you exhaust everything about apples, and the same about infinity? If not, then is it necessary for thoughts to exhaust everything about some property or object to still be about those properties or objects? Even though our thoughts of apples might not exhaust everything that makes an apple an apple, apples still exist, right? So could it be the same case for infinity - that our thoughts about infinity don't necessarily need to infinite to be about infinity. It seems that is what thoughts are - a model of what it is that we are thinking about, and models don't exhaust what it is that is modeled, but still have a (causal) relationship with what is modeled.
With an idea, we compare, contrast, measure, in short: relate it to other things. Every part of this process derives some portion of understanding.
I know there's a difference between "average velocity" and "instantaneous velocity" that's evidently obvious. However, "instaneous velocity" is still an average. It's just a different average from what is called the "average velocity". Here is how the page defines "instantaneous velocity":
"It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero."
Notice the word "average" there? I don't see why this is so difficult for you. Any determination of velocity, is necessarily some type of average due to the nature of time. It requires determining the difference between two distinct sets of circumstances to produce one result, called "the velocity". That is an averaging, coming up with one description from the two, you take an average between the two.
Do you see how this notion of infinity is inconsistent with constructivism? The bigger number referred to is not something which the human mind could ever apprehend, therefore it is beyond the capacity of understanding through constructivist principles. It's something which is simply stipulated, but never grasped therefore outside the range of intelligibility for constructivism, just like the "God" of the ontological argument, which is an inverted type of the same principle. "That than which nothing greater can be imagined", is a stipulation, which by the very nature of the stipulation cannot be grasped, because we can always imagine something greater. The same thing is the case with your "bigger" number, you are simply stipulating that no matter how big a number you can come up with, there's a bigger. The number you come up with is within the grasp of the mind, and comprehensible, the bigger number is always outside the grasp of the mind, therefore not comprehensible, and outside the principles of constructivism.
In short, you are suggesting that there is something (a number) which we can understand, which is outside of our range of understanding. The principle you propose is actually unintelligible.
Right, so the question that follows is: what happened so that we generally rejected constructivism? And what are the philosophical costs of having done so?
I thought @Banno had moved towards solving his own problem and was about to congratulate him. Have I got this wrong?
So,it's a shared idea? What's the problem? Don't we all have (more or less, that is sufficiently) the same experience and understanding of quantity?
I don't see how '1' could refer to '1' ; it is the referrer. not the referent, according to any logic that makes sense to me. I don't see a problem with '1' referring to 1; the apparent problem arises because we want to reify 1, and be able to say just what it is. I think it's just a conceptual illusion of substance.
I would rather leave the word "object" out here, because it invites reification. Better just say "abstraction" instead.
If by "abstract object" you mean a platonic object, then I think you are drawing a long bow in claiming it is the "prevailing view" among mathematicians. Even if we granted that, though, this being a philosophy forum, the prevailing view among mathematicians would not be as relevant as the prevailing view among philosophers.
What do you think "abstract object" could mean outside the context of platonism?
You might find this article helpful.
I don't see how "1" can refer to 1 in the same simple way (let's say) "justice" or "beauty" refer to justice or beauty. There's almost no information in the phrase: "1" refers to 1. In my mind, the second 1 is begging for quotation marks. Translated from numeralese, it would read: The numeral one, barring some preliminary qualification, refers only to the numeral one.
Clearer to say "1" refers (not to 1 but) to the concept of the singular or of the first.
At any rate, I don't find all of this very useful or interesting. It's obvious enough that "1" refers to something.
Yes, I agree that it is clearer; but I would tend to say that one is the concept of the singular or the first. In the usual philosophy of language 'x' ( any 'x') refers to x. If x is an idea then it can be expressed like this: 'x' refers to
If I think about justice, for example, I can ask whether justice is an idea or an activity or action. So, 'justice' refers to justice, where justice is thought of as an activity, but I might write: 'justice' refers to
That makes sense.
If I compare:
"Tree" refers to tree.
and
"That tree" refers to that tree.
- the first phrase makes almost no sense to me and the second phrase is fine. (Maybe the first refers to the concept of a tree, but it isn't really clear.)
Repeated with "1":
"1" refers to 1.
and
"That 1" refers to that 1.
Neither of these make much sense to me.
How about "'that one' refers to that one"? Would saying "'one' refers to one" be different than saying "'1' refers to 1"?
I think I see how it works.
Sure. But that fact that someone can add one to any number does.
Yeah. See how delta-t becomes zero? So your average is a division by zero.
I don't know how to help you see the error you have made; my replying to you just backs you further into a corner.
Quoting Metaphysician Undercover
But that's not right; mathematicians, even those in primary school, do apprehend infinity in their considerations. There are whole books about it.
Further it is clear that infinity of one sort or another is easily constructed from a few simple considerations.
And i think that is an end to this discussion.
:up:
I've come to realise that what I have been calling constructivist maths is not quite what is more generally called constructivist maths.
I reached a conclusion fairly early, and it seems that the talk of extension and intension in Wittgenstein’s Philosophy of Mathematics was misleading - whether in being eccentric or just wrong, I'm not sure.
It's something I will probably come abck to, but for now I am content.
It was a good old puzzler. Had me stumped.
What? Delta-t doesn't become zero. It "approaches zero". Can you not understand the significant difference between approaching something and becoming it? If delta-t was actually zero, it would render the whole formula as nonsensical.
Quoting Banno
Sure, we apprehend infinity, but not necessarily in that way. That way is inconsistent with constructivism, as I explained.
Quoting Banno
Yes, it seems to be approaching zero. But your capacity to argue a point is already at zero it seems.
Correct. Most people understand that.
I'd answer that with simplicity sake.
Quoting frank
I'd answer that with significant misunderstanding, as demonstrated by Banno.
Quoting Baden
Banno appears to be a lost soul.
SO do you agree with Meta that there is no such thing as instantaneous velocity?
For all practical purposes yes, we calculate instantaneous velocity. Actually instantaneous? Of course not. Most people can plainly see that that wouldn't make any sense.
It's just how fast something is going at some particular time. It's a basic bit of physics.
So, no.
Wittgenstein talks of a picture having one enthralled; unable to see something in a different way. Hence the duck-rabbit and such. This is perhaps a case in point.
So a physicist using classical mechanics would say that an object has only one location at an instant, but that it can have both a velocity and an acceleration.
Meta has an idea - Aristotelian, perhaps, that since an object can't go anywhere in an instant, it can't have a velocity.
That just sounds like a bit of Zeno-ian silliness!
Quoting Janus
Yeah, it is.
It leads to the confused reply he gave to my OP: Quoting Metaphysician Undercover
There's a certain coherence in what he is saying; and it is said with such conviction. It also seems to me to be a very similar to the misapprehension he had in @Sam26's discussion of rules.
"From a conceptual point of view, instantaneous velocity is a limit: if you compute the average velocity (?x/?t) for every smaller values of ?t, you will see that it nicely converges to a value: this is the instantaneous velocity. From an experimental point of view, this is unreachable."
Physics stackexchange
What is not reasonable is to call any sort of velocity "instantaneous velocity" because any velocity requires a period of time, and "instant" implies a point in time. So that phrase is really self-contradicting, oxymoronic. Because physicists use that saying, it gives people like Banno the impression that they can actually figure out what the velocity of something is, at a point in time, when they really can't. So it's a misleading (deceptive) use of words.
Quoting Banno
I would say that this is obviously contradictory. Movement is change of location. Velocity is an attribute of movement. Therefore it is impossible that an object could have one location, and also velocity.
Quoting Banno
Yes, that is my idea, it's known as conformance with the law of non-contradiction. You might call it an Aristotelian principle, I would prefer to call it common sense. We normally reject contradiction out of common sense.
Quoting Banno
If you are going to argue that language use is a matter of following rules, then it makes sense that you would actually follow the well known fundamental rules, in your argumentation. Otherwise it's hypocrisy which actually shows the falsity of what you re saying. So if we cannot adhere to the fundamental rules, the law of identity, the law of non-contradiction in our discussions of mathematical axioms, what's the point in saying that language use is a matter of following rules when actual usage demonstrates otherwise?
Quoting Banno
Contradiction and equivocation are abundant in mathematical systems. It's very clear that rigorous philosophical discipline has not been adhered to by those who have dreamed up the axioms. It appears like the axioms are designed to hide the problems which we have in understanding the nature of physical existence (such as Zeno paradoxes), rather than to expose these problems so that we can work on resolving them. The hiding of the problems creates the illusion that they have been resolved, which many people seem to believe as reality. But issues like the uncertainty principle demonstrate very clearly that the problems have not been resolved.
A secondary type of problem has now emerged. This is an even worse condition than the original problem, which is our inability to understand these aspects of physical reality. Since many people believe that these artists, the mathemagicians who have dreamed up the axioms that are capable of covering up the problems, have actually solved the problems, they falsely conclude that there are aspects of physical reality demonstrated by QM, which are incomprehensible. Instead of accepting the fact that the mathematical axioms which are employed are stacked with logical flaws, and this is why certain aspects of physical reality appear incomprehensible, they will defend the mathematical axioms to no end, and argue that this is just the way nature is, certain aspects of physical reality are fundamentally incomprehensible. For example, there is a commonly expressed attitude that the uncertainty of the uncertainty principle is a fundamental feature of physical reality, rather than a deprivation of the mathematical principles employed. Do you see how wrong this attitude is?
It does refer to 1. 1 = 1 but it's not how it's written, it's how it's concieved.
Physics can differentiate the two at time t by different motion vectors, speed and direction; by momentum too for that matter.
If you can't, then you're missing something.
Simple school physics could plot out the different speeds at different times throughout the scenarios, and see acceleration/deceleration (change in speed) over time; the former scenario would be a bit boring.
If you can't, then you're still missing something.
Gravity expressed as acceleration (the equivalence principle): at time t, Earth gravity is a downward acceleration that we're subject to.
Without differential (and integral) calculus, physics would be impoverished, it's proven in action, so our philosophical musings best account for this, or we'd be missing something.
[quote=Asimov (1941, 1990)]So the universe is not quite as you thought it was. You'd better rearrange your beliefs, then. Because you certainly can't rearrange the universe.[/quote]
No, that just implies you can add one - a finite value - to any number - another finite value. So where does one get the notion of infinity from when you are starting somewhere in using numbers to count and then simply adding one to where you started.
The notion of infinity comes when contemplating things not just with no ending, but no beginning as well - like a circle or the visual feedback loop that you observe when looking in mirrors positioned in opposite sides of the room.
You keep confusing potential infinity with actual infinity.
When you glance at your speedometer and it reads 60 mph, indeed that is based on an approximation made over a small interval of time. So you do have a point, although a rather insignificant one. "Instantaneous" velocity or speed is a shorthand for a limit process. What single word would you suggest be used in this context, rather than instantaneous?
6/3=2
Again, a major problem in philosophical discussions is exhibited. :sad:
Yes, but you seem to be ignoring what I said. If what you and I both said is true, then how do we reconcile our opposing, but true, viewpoints? I was hoping for something like this but while pointing out the problem you failed in trying to solve it.
So basically, if you have 6 apples and three people, then the number of apples divides equally, but try dividing one apple evenly among three people.
Ok, what now?
Quoting Harry Hindu
Well, no.
? = 3.14159...
1/3 = 0.333...
Sure, the righthand side has unending digits, but don't confuse the representation and the number.
Quoting jgill
(y)
It's all contextual.
Similarly with differential calculus; if the plot has a sharp turn or just one point (which would have no context), then it's not differentiable, which would represent something we don't really see much in the world.
The problem is that there is no such thing as motion at time t. You might say that there is motion at an extended duration of time, and infer that because of this there would be motion at any given point during that time duration; but that would be a faulty inference. It would be like saying that at any point on a line segment, there is a line, just because we have assumed a line which goes through that point. But there is no line at any point, just like there is no motion at any point in time even though we assume that motion passes through that point. The two, motion and point in time, are incompatible, just like point and line are incompatible.
Quoting jgill
How about just calling it "velocity"? We know that "velocity" implies an average over a period of time, just like "instantaneous velocity" implies an average over a period of time. The method for figuring out the average which is called "instantaneous velocity" is just more sophisticated than the old fashioned way of figuring out "average velocity", so it may give us a more accurate or precise determination of the same thing, "the velocity". Nevertheless, the two are just different formulas for giving us the same thing "velocity". So use of the word "instantaneous" is rather deceptive, it does not properly indicate what the formula gives us..
I've never read much of Harry's stuff (on the suspicion that more is less) but, for the second time this weekend, I do applaud him for going against the flow, and I must say I can't understand how people would so miss the point, and would take the above rhetorical question as anything but a defense of mathematical practice against philosophical over-thinking. He was just saying, see how the fact that we can divide one by 3 despite the potentially infinite recurring decimal (Achilles can catch up) means we don't have to (in this case anyway) take infinity as a thing.
Wasn't he?
Velocity or momentum or some such vectors (at t) depend thereupon.
It's not like we have something appearing and vanishing at t, whether talking averages or differential calculus.
How/can you differentiate things at t in the two mentioned scenarios...?
True enough, if taking a photo of a moving object - which has the effect of freezing the motion. We use time = t in lots of formulae, and make accurate predictions. But in everyday affairs we experience time more as intervals, although we say things like "I'll meet you at three".
Is time flowing at time = t? I suspect it is.
My point was accuracy of statement. Philosophical overthinking seems normal on this forum. :cool:
Thanks for the excellent clarification.
When we use a calculator to divide 1 by 3, we get 0.333...
Try as we might, we can never put the 1 back together again with evenly divided thirds in the calculator, because it would require you to enter an infinite amount of 3's after the decimal, yet we end up giving 's divided apple to just one person, is that person missing any of the apple?
We agree that the location does not change at an instant.
Where we disagree is that there are those amongst us who are happy to ascribe a velocity at a particular time, and those who are not.
What is hard to see is how those who do not ascribe a velocity at a particular time can do any basic mechanics.
It's the 0.9999... = 1 denialists, hard at work again.
Time t has no context. If you say "time t", "time" is said at a different time from when "t" is said, because time is passing. So time t covers a duration of time. By the time you say "now" it's in the past. Talking about "time t" is already, by that fact, a removed from context; context being real existence in passing time. It is impossible to have a time t which is not a removal from context. That's the problem here, time t is an ideal which is not consistent with temporal existence as we know it.
Quoting jorndoe
The problem is that "time t" is not real, it's an ideal. And Banno wants to understand these things without assuming Platonism, so we must reject such ideals, as not reality. So asking about how we might differentiate things at t is nonsense because "t" doesn't refer to anything real.
Now you've jinxed it. :D
Actually we haven't gotten to these questions yet. As is evident in the prior post, I think "a particular time" is an ideal, which on it's own is without any real validity. What validates it is a reference to something.
Quoting Banno
I do a lot of basics mechanics. Complex mathematics is not required for basic mechanics. In fact, mathematics is generally not required for mechanics at all. Fancy that.
Sure it does, especially how we're talking about it here, other times, events, occurrences, you name it.
Actually, I'm not sure it's coherent to go all out context-free here.
Quoting Metaphysician Undercover
Excellent.
I'm going to quote you on that next time I'm late for a meeting with my boss.
Indeed. But look to who is on which "side".
How do you work out the velocity at t2 if the velocity at t1 is always zero? :rofl:
Obviously, since at 1second, the mass cannot move, its velocity will be zero.
But then, of course, I'm assuming an acceleration of 9.8m/s/s. And that's gotta be wrong, since an object can't accelerate without moving.
SO come on, help us re-write the physics texts.
I don't remember the acceleration of gravity. I do remember that there's an infinite converging progression involved in answering your question.
What do you conceptually commit yourself to if you embrace an answer that can't be witnessed experimentally? The evidence is entirely intellectual.
More strawman. Why?
WTF?
Just like "t1" is an ideal, so is "t2". I thought you rejected Platonism? Do you believe in Einsteinian relativity?
In arithmetics, it doesn't really mean much.
In some cases, in calculus, it can.
Best not conflate, the angle matters, context matters.
If we only want to speak of intervals, non-zero durations, then what about the starts and ends thereof?
Are we going to toss it all out...? :o
Anyway, successful tested-and-tried application speaks for itself.
That's another aspect of the very same problem. I'm not suggesting that we toss any of these things out, only that we recognize that in practise all such determinations are less than ideal. Then we might be inspired to look for solutions to the problems which result from using such deficient principles, instead of just assuming that the mathematicians have already discovered the ideals.
Maths aren't things we find? Fractals and the Mandelbrot set seem to disagree with this. We cannot "do" infinity by definition. Finite beings cannot create infinities. Only infinite beings can create infinities. Which is one of my arguments for God, ironically.
1) Infinities exist.
2) Finite entities cannot create infinities by definition, because finite beings are limited, infinities are unlimited.
3) Infinities are caused by infinite beings.
4) Infinite beings exist. Ergo,
5) God exists.
Thanks!
No, I didn't prove that at all. That's totally a non-sequitur. So, just because I use language doesn't mean I created English does it? Obviously not. You can talk about infinities all you want to. But you cannot produce one. For you to produce something that would go on forever, you yourself would have to live forever to do it. That's the whole point.
All you need to do is define "infinity" in such a way that you can produce them, and voila, you can produce infinities. It's a very simple trick which the mathemagicians do with their axioms. However, we need to respect the fact that when they talk about infinities they are not talking about the same thing as you, when you talk about infinity.
Then 1 does refer to things, like velocity and time. Glad to see that you finally see that I made sense, Banno.
Now, the question is, does speed and time exist as something other than a concept, or as a potential, not an actual, like infinity?
What is an instant? Think about what an instant looks like to you and then what an instant looks like to a cold-blooded lethargic lizard. That cat is moving at a relative speed (velocity) of 15 mph to your eyes, but to the lizard, it's movement was instantaneous.
In talking about velocity, we are talking about the relationship between a certain change in location relative to something else during a certain time. Velocity in miles per hour is how many miles (the relative) something moved during one hour.
Now, if the lethargic lizard could measure velocity, while the velocity of the cat would be instantaneous to the lizard and take time to you, the change in location of the cat relative to the length of a mile will still be proportionally the same. So while our perspectives of time and velocity may be different, the proportional relationships stay the same.
2. You divide the set by 1, equating 40 for Y.
3. Order is lost in simple division, treating all different cards as label 1.
4. Repeat step 1.
5. You divide the set by 2, equating 20 for X.
6. Order is implied for X has 20 specific cards out of 40.
7. Order is implied for Y.
8. 1 is not powerful enough a number.
2/40 can be a number of different cuts; at step 6 there are many possible sets of cards in X's control, and thus 2/40 = 20 is wrong; overly simple. You can try the same method with all labelled 1, and get the same result. There is an error at step 2.
40 cards are not the same as 40 1s. 1s seem to blend, per se, to make 2. No matter what, if you have 2 cards. That's card 1 and card 2, never a single card
So 1 + 1 does not equal 2, but rather (1 1), which can be said to be 2, but, following on from 2, is stupid, it's just a reference to (1 1), follow on from (1 1).
End.
No. You can model a system that produces infinities. But you cannot actually create an abstract thing that is infinite. That's a different thing. I mean technically, replaying a video game each time will produce an infinite number of different conditions in the game, being different each time. The game isn't infinite. That's the system that is modeling infinity, not the actual thing. Simulation vs simulator. Not the same thing. I keep having to repeat this difference.
I'll take that under consideration since you obviously have an in-depth knowledge of the subject. :roll:
2. There is a refill ammo box.
3. I place a camera on a wall, and then refill ammo and continue to place cameras.
4. I can do this until: the game is impossible to play, there is no space for cameras or I destroy the ammo box.
A model of infinity is impossible, but not an infinite practice.
The walls fill up, the game costs, requiring that I put in effort to run the program infinitely. Riding a eco cycle to supply energy to my console, removing and replacing cameras on a wall, and THEN I could go on infinitely; however, it's a different infinity than the original. I'm not, 'infinitely placing the cameras', I'm infinitely replacing cameras (which means I need to place X amount of cameras).
A great cog is required for any practice of infinity.
Yeah, I think we've met on some other threads with similar subjects. Now I think you're beginning to catch on. It's just a matter of analyzing the axioms, in order to understand what they actually mean. I would recommend this as a revealing practise for any philosopher. Mathematicians on the other hand seem to be disinterested, being more inclined to take the axioms for granted as if they are some sort of eternal truths.
Mathematics evolved over millennia and foundations are fairly recent. Most practicing mathematicians, especially those in classical mathematics, just do the math they are interested in and avoid arguments over the axioms that lie at the base of foundations. Of course, analytic philosophers, set theorists and other math people can be heavily involved in foundations, and keenly feel perturbations in that structure that would go unnoticed by the rest of us.
For me, arguments in transfinite mathematics seem far too abstract, but for others they may represent the soul of the subject. Personally, doing minor research in classical complex analysis I've never needed to go transfinite. But others in what is called modern or "soft" analysis have used debatable axioms like the Axiom of Choice for their investigative results.
If you are a person who feels strongly that the axiomatic structure of math contains flaws, the go for it. There's room for everyone. :cool:
You ought to recognize this as contradictory. The foundations are what something is built upon, and therefore cannot be something recent when the thing has been around for millennia. So this statement implies that you misunderstand what the foundations of mathematics really are, interpreting something recent as the foundations, when this really cannot be "the foundations" which must refer to what the thing is built on. Take a look at Banno's op, there is a reference to "counting", I suggest you'll find the foundations of mathematics here. But counting has two very distinct purposes, one is to determine a number of things, quantity, and the other is to determine an order of things, priority.
Because of these distinct purposes, we have ambiguity and the potential for equivocation right at the basic, most fundamental principles of mathematics. So "number one" refers to an individual, a particular, as distinct from others, for the purpose of counting, but it also refers to the first in terms of priority. With the introduction of zero, and negative integers, "one" has lost its status as the first, so we need other principles to understand the concept of priority. Where are these principles of priority? It appears like modern mathematics gives us no principles of order, having given priority (importance) to quantity at the cost of sacrificing order. The result, modern mathematics is a disorderly mess.
Quoting Metaphysician Undercover
I know. It's a tragedy that requires competent philosophical guidance. Thanks for being there when we need you! :scream:
Wait, hang on. Does repeat refer?
"The" does not refer to anything. Is this a problem?
My physics teacher used to say 'a number means nothing until you state your units.'
The solution is that 'the' does not refer until you say "solution".
"1" does not refer. "1 sugar in my coffee" refers as much as anything because the units have been specified.
Hurrah for physics.
No, few people listen to any philosophers, and that's a tragedy in itself. So we have a double tragedy, philosophical guidance is needed, but it's not heeded. My existence is irrelevant.
This seems to be primarily an amateurs' forum - and I don't mean this in a pejorative way - in that few if any make their living as professional philosophers (probably requiring graduate degrees). Your ideas on the foundations of mathematics might receive a more serious scrutiny were you to post them on a site like https://www.google.com/search?client=firefox-b-1-d&q=math+stackexchange. Yoiu might find some there who would agree with you. Just a thought. :cool:
I wonder if you know of any good forums for psychology or sociology...Thanks!
https://thephilosophyforum.com/discussion/comment/408210
Start with "/"
Iterate: //
And again: ///
And again: ////
So you get ///////////////////...
Partition each step: /, //, ///, ////,...
These partitions are sets
{/}, {//}, {///}, {////},...
and they are represented in Arabic numerals as
1, 2, 3, 4,...
The initial / need not be anything other than a concept of something or nothing.
In mathematics it can be the null set.
Counting infinity has nothing to do with time. An infinity of numbers does not require time to exist. They exist conceptually as a set.
In a sense. However, as a mathematical analyst, when I iterate w=f(z)=z+1, starting with z=1, the process is unbounded and hence is said to diverge to infinity. In a computer program each iterative step requires a tiny amount of time, so time is tied in with this notion of counting infinity, although in theory the pace is arbitrary. Is it possible to think of a process that counts to infinity and does not require a step-by-step procedural? Certainly the concept of the set of counting (natural) numbers as a theoretical entity is not tied to time. :chin:
It requires an infinity of time to count an infinity of numbers, so "counting infinity" does have something to do with time.
Hm. It requires a sequence, sure. But a sequence is not the same as time.
Consider a Koch Snowflake, which has a finite area yet an infinite perimeter...
...and in considering it, one finds oneself as it were, standing outside of the iterative process that creates the flake; one understands the flake despite not having performed every iteration.
Unless ons is @Metaphysician Undercover.
Understanding is not the same thing as counting.
You cannot count an infinity of numbers, so time has got nothing to do with it.
Counting is a temporal process. Two comes after one. Three comes after two. You cannot remove the temporal aspect of counting, to claim that time is irrelevant to counting. It is essential. Try counting when four comes before three. It doesn't work.
Other problems ar3e caused by considering that axioms and theorems are two distinct entities, when in fact one runs into the other. ie a theorem can be used as an axiom. or 2+2=4 can be considered to be axiomatic.
Also maths might be less confusing to philosophers if they stuck to numbers in base 1 instead of base 10.
Pure maths is entirely abstract. You are conflating pure maths with applied maths. Numbers in pure maths do not require counting.
Sure, but we were talking about counting, not pure maths. The contested statement was:
Quoting EnPassant
But can be thought of as correlating with linear time, each step separated from the next by a short period of time. Maybe an isometry, but that's not quite right. Unimportant except for philosophers. :smile:
However, since nobody is constructing the sequence ad infinitum, it can't be said to go on forever. So the question becomes how a constructionist can justify a concept of infinity if it's never constructed. Otherwise, one is granting the Platoniist's argument that the sequence already exists.
So in what sense does it mean to say that 1,2,3... goes on ad infinitum?
Wouldn't that require time to be discrete?
Infinity just means 'without end'.
We can do that, but does that work for construction? You're saying imagine a number bigger than any number the rule can construct.
Sure, but then you have the problem of how the .333 repeats forever. It can't already exist on the pain of Platonism, nor can it be generated by a rule.
It seems like you're having to step outside the rule to add something. And what is that? The idea of the rule repeating forever.
So then "infinity" means a rule that never ends, but can't be generated.
Cantor uses Aleph Null to count infinities. One can count an infinity conceptually, without time. How much time is there between the digits of pi? Likewise with the empty question 'What came before the beginning of time?' The real question is "What gives rise to time?" or "On what necessary condition is the world/universe contingent?" It is really an ontological question.
In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together.
That's not counting though. Anyone can make up a new definition of "counting", and use that definition to make whatever conclusion one wants to make about infinity. But that conclusion would be irrelevant to what "counting" really means to the rest of us. So if Cantor turned "counting" into some sort of abstract concept which has nothing do with the act of counting, as we know it, I don't see how that's relevant. You are just arguing through equivocation.
The difference is really semantic. Counting is about associating a number with an object; 1 orange, 2 apples etc. But Cantor counts numbers with numbers by associating numbers with other numbers. In this way Cantor associates/counts the rational numbers with integers and comes to the conclusion that there are enough integers to count the rationals.
So how does a constructionist handle such a number? Do they deny that the set of all numbers is properly mathematical?
Kummer, Cantor's arch enemy, was a kind of constructionist and denied the reality of real numbers. I guess they just don't agree. The question here is What does 'real' mean when we are talking about (what seem to be) abstractions? What does 'exist' mean in the context of numbers existing?
Yes the problem is semantic, that's what I said. If you give "counting" whatever meaning you want, you can do whatever you want with it. So Cantor is not really counting as "counting" is commonly used, because number is a concept, and not a thing which can be counted. It's only by making numbers into mathematical objects (Platonism) that Cantor can count numbers.
Setting side those never ending debates, what does it mean for a constructionist to be able to offer a proof for any conjecture involving an infinite sequence, such as any number greater than two is the sum of two primes?
It seems to me that a pure constructionist cannot even admit that there are an infinity of natural numbers. Induction proves that there are and many mathematical proofs rely on induction. But this comes back to what we mean by 'exist' in relation to numbers in a Platonic sense. What does 'exist' mean?
One unresolved question in philosophy is why there is something rather than nothing. We don't know but we know there is something. This necessary something that is, before all created things, is what is, eternally. This eternal substance is existence. It is not that this necessary something has the property 'existence' it [I]is[/i] existence because existence cannot be a property. So, if numbers exist, they must be intrinsic to existence. And since it takes Mind for numbers to exist, existence must be Mind, if numbers are in existence. The only eternal mind in which numbers can exist is God's Mind.
What all this means is that existence, mind, and God are three names for the same thing.
In this context I am using the word 'existence' to mean that which necessarily is.
I don't know, but it's difficult to say that math is entirely made-up when it's so useful in scientific theories. Quantities of things exist, so does topography and function.
More or less true in set theory, a particular branch of mathematics. My area was complex analysis and when I deal with the concept of infinity it is in the sense of unboundedness of sequences or processes. :cool:
Here is a thought. Write the squares of numbers like this-
1 squared = 1
2 squared = 4
3 squared = 9 etc.
Now, you can plot this sequence of squares on a graph as a quadratic curve, the curve of x^2.
The question is, how can a flat piece of paper receive this concept of squared numbers so faithfully? How is it that it is possible to translate a thought about numbers onto a graph in flat space?
This can only be possible if there is a natural correspondence between mind and space. If mind and space were utterly different it would not be possible to create an image of mathematical ideas on a flat space. But if there is a natural correspondence between mind and space what is it? The only common factor I can think of is mathematics. That is, mind and space must be intrinsically mathematical.
Quoting jgill
Yes, but the limit can be defined independently of time.
Of course it can. I merely mentioned a kind of isometry between iteration and time. I deal with infinite sequences whenever I dabble with research, and I rarely consider a correlation with the passage of time.
Forever...
Not all forevers are temporal. A line does not require time. Quoting Metaphysician Undercover
Have you never seen a number line?
A number line is an irrational conflation of two incompatible terms, discrete numbers, and a continuous line, that's why the idea creates so many problems. If there is a line which extends "forever" it is spatial. If the number line is supposed to count forever, then "forever" is temporal. But the simple line which extends for ever doesn't count anything.
So this sort of thing must be a real bitch for you...
It's quite boring, if that's what you mean.
exist?
Have we left the question of Wittty's finitism behind?
and
Quoting Banno
It makes doing mathematics like walking across a minefield! :fear:
Here's a thought. Draw the X axis. The segment between 0 and 1 is a physical extension in space. This segment contains an infinity of dimensionless points: ie points of zero dimension. But if you set down an infinity of these zeros side by side, you get 1 unit of length. The implication is that 0 x [math]\infty[/math] = 1.
So essentially any number would not refer to anything either? If so what does zero refer to? What differentiates 1 from 0?
It just means there are conundrums lurking whether you embrace constructivism or reject it.
What you do with one is not the same as what you do with zero.
Which gives it a value higher than nothing?
Needless to write 0 and 0 is 0 while 1 and 1 is 2. Has to have some value?
Yep. Your point?
Maybe I haven't been following the discussion but, therefore "1" has to refer to something? Not "anything" I suppose but it certainly doesnt refer to nothing as in 0.
One is greater than zero.
The only way one could be greater than zero is if the thing one refers to is greater than the thing zero rferes to...
Hence one must refer to something.
Is that your argument?
Sort of. Folks call zero a placeholder. It's the youngest of all numeric concepts, if I'm not mistaken.
If we are referencing something, using numbers. Zero does refer to something, rather lack of it. One is one unit, item, whatever the thing is. Two is two so on and so forth. Right?
Well transcendental arguments are notorious in that they rely on "the only way...". My point is that one is greater than zero not because of what it refers to, but because of the way we treat it. Would you rather one dollar or zero dollars? And hence that "the only way..." fails because there are other ways.
My argument, so far as I have one, is that the extension of "one" - the bunch of things to which "one" might refer - is each and every individual; and hence, the extension of "one" is anything. And if that is so, then one is not differentiable from anything else; it makes no sense to talk of the extension of "one". The corollary is that since numbers are all built form one, none of them refer.
But you seem at the end of your last post to offer another argument; that "zero" refers to nothing, and hence is another example of a number that has no referent.
It's used as an adjective, so it's like "blue" or "fast." I have one elephant. It's modifying a noun. So what's the extension of an adjective?
It can also be a noun: "One is the loneliest number." The reference us a number.
Hence it makes sense to talk about blue things because there are things that are not blue; there is stuff that is not part of the extension of "blue"
But if the extension of "one" is each and every individual thing, then there is nothing outside of that extension.
Yes that's a pun on "nothing".
Blue can be used as a noun, as in "cobalt blue". We're contrasting a hue or shade with others.
When "blue" is used as an adjective, it's reference doesn't appear to be a spatiotemporal object. Maybe it's not the kind of thing that has a reference.
Would @fdrake know?
You are asking fdrake if he knows if you agree with the standard definition of extension in logic?
The proper understanding of math derives all numbers, mathematical operators, and all the other mathematics, from the starting symbol 0.
Then as by rules most parsimonious, there would be an ordering of mathematical structures, in regards to the zero.
The start would be 0. Then to derive the 1 from the 0, the 0 is rewritten as a 1. The rewrite principle is like when the same information can be rewritten to harddisk, and to a dvd, and to RAM, and it's still the same information, but rewritten in a different form.
So the 1 is essentially a rewrite of the 0. Now we also get the boolean operator, because 1 being a rewrite of 0, means they have boolean interchangeability.
Now we have the mathematical structures of 0, 01, 10, 00, 11
0, 01, 00, and 10 would all total 0, because of the boolean operator making the 01 and 10 also total zero. But there is no boolean operator for 11, making it not total zero, which is totally uncool.
Therefore logic must respond to make it total 0 zero again. So logic responds to the 11 with a 00. And all is parsimoniously total zero again, and we get additional mathematical structures. And so on.
And it should be shown to be the case that in physics, the physical 0 and 1 in isolation, would have boolean properties, that this would be some kind of phenomenon of physics.
And the main laws of physics, and constants, would be apparent in the ordering of the mathematical structures in respect to the zero.
And the human mind, and the DNA system, and the universe, would all be shown to have the same fundamental mathematical ordering. But no the sequence of CATG, or where the planets and stars are, and what pictures someone dreams of. That is not fundamental.
I'd call it a video loop. What does it have to do with a number line, or counting? A loop is not a line, and if it's counting anything it's counting the same things over and over again.
:cool:
The extension of a property is the collection of objects which satisfies the property. "is an object on my table" has extension "my laptop, a half litre coffee mug, a heat mat, a candle holder, a plastic water jug, a 2 factor id device, an unplugged microphone, a computer mouse and 2 boxes of oral nicotine pouches/snus".
https://www.britannica.com/topic/intension
In terms of your discussion with @Banno, extension and intension are different ideas in general than referent, except maybe in the case where you're already dealing with a word or phrase which is being used to refer, like the proper noun "frank", with my intension you.
1 can count the 0. 'One zero'.
Huh. Suppose it can. :D
You can create the number line with the null set. Let {0} = the null set:
{0}
{0}{0}
{0}{0}{0}
{0}{0}{0}{0}...etc
= 1, 2, 3, 4...:
And then come the fractions . . .
'One infinity'.
One anything. One {0} and off you go...
Numbers are made by iteration and partition.
Start with- "/", iterate: "//" and so on: "///////////////..."
Partition each step:
/, //, ///,...
= {/}, {//}, {///},...
In Arabic numerals-
1, 2, 3,...
Actually not quite. You can't have one infinity divided by one zero. Or if you can, the mathematical universe goes all indeterminate on you.