Physics tells us that even bricks are nothing more than probability waves smeared across the universe — fishfry
If you don't believe in sets, why go to the trouble of explaining why you don't believe in the empty set? I wonder if that shows that you haven't thought your idea through. Why bother to argue about the lack of elements, when you don't even believe in sets that are chock-full of elements? — fishfry
Nobody has claimed sets have "real" existence, whatever that is — fishfry
I could easily take you down the rabbit hole of your own words. Is an electron "real?" How about a quark? How about a string? How about a loop? And for that matter, how about a brick? Are there bricks? When we closely examine a brick we see a chemical compound made of molecules, which are made of atoms, which contain protons, neutrons, and electrons, which themselves are nothing more than probability waves smeared across the universe. — fishfry
Do you believe in the existence of bricks? Physics tells us that even bricks are nothing more than probability waves smeared across the universe. — fishfry
Do you deny science along with math? — fishfry
some argue that the real numbers are not truly continuous — aletheist
The Peirceans who are not on this forum, for starters; but it goes back at least as far as Aristotle, who recognized that numbers of any kind are intrinsically discrete, rather than continuous. The key word here is "truly"; I have acknowledged that the real numbers are an adequate model of continuity for most mathematical and practical purposes. Nevertheless, conceptually a line is not composed of points, a surface is not composed of lines or points, and a solid is not composed of surfaces or lines or points. Instead, the parts of a line are one-dimensional lines, the parts of a surface are two-dimensional surfaces, and the parts of a solid are three-dimensional solids. Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts.Who argues that, exactly, besides the Peirceans on this forum? — fishfry
Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts. — aletheist
A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts.The vocabulary of sets lets you phrase all these concepts already. "Parts of a line? They're finite intersections of its interval subsets which have cardinality greater than than 1". — fdrake
A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist
Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line.Let's say I have some "True continuity" X. Like a line X=(0,1). — fdrake
No, again, a line is not composed of points corresponding to numbers. We can only mark them on (not in) a truly continuous line. They then serve as arbitrary and artificial limits/connections between distinct portions/parts.So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. — fdrake
Not at all; again, set theory can be quite useful as the basis for an approximate model of continuity. However, it cannot serve as the basis for true continuity, because it requires discreteness at the outset.If not, do you reject sets as a concept? — fdrake
A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist
The Peirceans who are not on this forum, for starters; — aletheist
Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line.Did Aristotle reject the notion of an instant of time? Or did Peirce? — fishfry
Line figures, surfaces, and solids can be understood in geometry as truly continuous. We use points to model and analyze them, but they are not composed of points.Do you know of any current mathematical objects that behave more like true continuity in your view? — fdrake
Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line. — aletheist
I am not sure whether it can actually work out that way, but that was the idea, if I understand him correctly — SophistiCat
Let's say I have some "True continuity" X. Like a line X=(0,1). Let's say I can take "parts" of it in the above manner; arbitrary subintervals. (0,a), (a-1/n,1) are subintervals for any a less than 1 and greater than 1/n. Since they're parts of a true continuity, the true continuity ensures the existence of their intersection; limit the process of intersection over n and get the limit {a}. If I take the union of all such limits, since a was an arbitrary member of (0,1), it produces (0,1). So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. It's top down and bottom up at the same time.
Do you agree this process is legitimate? — fdrake
You then stipulate that any function ff defined on D→R∪DD→R∪D has some unique constant bb such that for all d∈Dd∈D f(d)=f(0)+dbf(d)=f(0)+db. There's also a property that d2=0d2=0 for all d∈Dd∈D. This looks like placing a family of infinitely small line segments around every point in R∪DR∪D. For functions, it gives a "tangent space" of a function at a point confined to infinitesimal neighbourhoods around it. It "smears out" the real line (and function values) into something that can't be element-wise disassembled in the same way (if you fix a point of RR, this corresponds to an infinitesimal neighbourhood around that point in R∪DR∪D) — fdrake
Well, you've arrived just where I have a problem, or, rather, where I'm unclear and confused, because I'm not arguing against any well-known fact, but rather I seem to be stuck in some misconception or misperception.
Taking your .10101010..., how long is it? How many zeros and ones? As many as there are counting numbers? Or more? ℵo or ℵ1? — tim wood
It's not a matter of what exists in reality, it's a matter of what is contradictory in principle. To say " I am going to talk about this object, but this object is not really an object, because there is zero of them", is blatant contradiction. To bring this expression out of contradiction we must amend it. I might say for instance, "I am going to talk about a type of object, of which there are none", or I might say "I am going to talk about a quantity, and this quantity is zero". But if I make the category mistake of conflating these two options to say "I am going to talk about this quantity, zero, as an object itself, and assert that there is none of these objects", then I contradict myself. — Metaphysician Undercover
Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line. — aletheist
It then follows from the first sentence that since the present is not an instant, there is no such thing as an instant at all.We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant. — Peirce, c. 1895
Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively.Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible. — Peirce, 1906
If you want to prove that ZFC is inconsistent you have to derive "false" using the rules of ZFC's logic. You can't do it using english language, as you are trying to do.
You can't be an art critic without looking at the paintings! — Mephist
You do recognize that "false" and "true" are assigned to the premises, not by what is determined by the logical system, (which is validity), don't you? Inconsistent, or contradictory premises, is not determined by the logic of the system. — Metaphysician Undercover
I criticize the axioms according to how they are expressed in English. If the fundamental axioms could not be expressed in English, or other natural languages, they would be meaningless. Terms need to be defined. — Metaphysician Undercover
But what type of meaning could this be, if when it is represented in English it is contradictory? — Metaphysician Undercover
We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant.
— Peirce, c. 1895 — aletheist
Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, — Peirce, 1906
Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively. — aletheist
The only reality that we can know is what we learn from experience. We formulate hypotheses to explain our experience (retroduction), work out their necessary consequences and make predictions accordingly (deduction), then test whether those predictions are corroborated or falsified by subsequent experience (induction).But isn't he conflating human experience with reality? — fishfry
What is reality? Perhaps there isn't any such thing at all. As I have repeatedly insisted, it is but a retroduction, a working hypothesis which we try, our one desperate forlorn hope of knowing anything. — Peirce, 1898
NO. "false" and "true" in first order logic (the logic used in ZFC) are purely SYNTACTICAL expressions. They are determined ONLY by the logic of the system. That's the way it works! — Mephist
In a formal logic system TERMS DON'T NEED TO BE DEFINED. That's why it is called "formal" logic. — Mephist
The proof of the theorem shows that a model always exists (if no contradiction is derivable) because it can be built using the strings of symbols of the formal language itself!
Probably that's the part that you strongly disagree with. But if you want to criticize the proof of Godel's completeness theorem, you should at least read it! That's what I meant by "looking at the paintings" before. — Mephist
The only reality that we can know is what we learn from experience. — aletheist
If you are asking me to accept the precepts of the system without judging them, then you are being unreasonable. — Metaphysician Undercover
A conclusion cannot be incompatible with the premise — Metaphysician Undercover
If you are merely talking about a set of rules by which symbols are related to each other, then there is no reason why we can't discuss these rules in plain English. I think that your refusal to discuss this in plain English is evidence that you know that there is deception within the system. So, either you discuss these rules in plain English or I level the accusation that you're attempting to hide deception behind your language. — Metaphysician Undercover
I don't see how your art analogy works for you. An individual can glance at a painting, and find it ugly without analyzing it, just by apprehending a few prominent features of it. — Metaphysician Undercover
It's completely unreasonable for you to say that I must accept the system's axioms in order to judge the system's axioms, when acceptance is dependent on judgement, and acceptance precludes the possibility of fair judgement. A conclusion cannot be incompatible with the premise, so if I accept the axioms, it is literally impossible for me to produce a judgement against them. Therefore you are being completely unreasonable. — Metaphysician Undercover
Were you like this when you learned to play chess? "This is the knight." "But no it's not REALLY a knight. Real knights don't make moves like that, they slay dragons and rescue damsels. I refuse to accept the rules of your game till you tell me what they mean outside of the game." — fishfry
You have to prove that assuming those axioms leads to a contradiction. — Mephist
You have to use the rules of logic to produce a sentence of the form "A and not A" (I am not sure if "true" and "false" are terms of first order logic, maybe I made a mistake before saying that you
have to derive "false"). — Mephist
The interpretation of the terms as sets (and then the meaning of the sentences) is a different issue.
You can argue that the terms that ZFC calls "sets" are not exactly correspondent to what we "intuitively" think to be sets, and a lot of people (even mathematicians) have this kind of objections to ZFC. But this is not about the consistency of the theory; this is about it's "meaning". — Mephist
That's not quite right. I learned how the game was played, then decided I didn't want to play it. The fact that it was a game, and the rules referred to nothing "real" probably made me think of it as a waste of time — Metaphysician Undercover
(https://www.brainyquote.com/quotes/bertrand_russell_402437)Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
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