Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again. — TonesInDeepFreeze

Do you know the law of identity? It states that a thing is the same as itself. It says nothing about equality or equivalence. That two things are equal is a human judgement.

And you said:

Rather, we infer they share all properties from having first proved that they are equal. — TonesInDeepFreeze

See, no strawman. You prove that they are equal (human judgement), then you infer from this, that they are the same. Let me put it simply, proving that they share one property, "are equal" does not prove that they share all properties. You need another premise, which states that equal things are the same thing. But we know that premise is false because we see all sorts of equal things (equal volume, equal weight, etc.) which do not make two things the same. Equality is not sufficient for a judgement of same.

We don't judge two things are equal. — TonesInDeepFreeze

You very clearly stated "having proved that they are equal". Therefore you do judge that they are equal, that's what proving is, providing the justification for judgement.

No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics. — TonesInDeepFreeze

The indiscernibility of identicals does not provide the principle required for substituting equal things. Things are judged to be equal not on the basis that they are indiscernible. Clearly, as altheist agrees, and what ought to be obvious to you, what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. Therefore it is false to claim that the principle of the indiscernibility of identicals supports such a substitution. It does not.

You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion. — TonesInDeepFreeze

Clearly mathematics has an extremely vague notion of identity, one not consistent with the law of identity, allowing that similar things which are judged to be equal may be substituted as if they are the same thing.

Good, and they also both denote the same object with "did," which is the relation of doing. — aletheist

I really can't see how a relation is an object. I think you are making things up as you go.

We already went over this with "Henry Fonda" and "the father of Peter Fonda." These signs both denote the same object despite signifying different interpretants because it happens to be a fact that Henry Fonda is the father of Peter Fonda. — aletheist

This explains nothing. Words like "did" signify something. But you insist instead, that they denote an object. But you also allow that they signify things as well, and denote objects at the same time. On top of this you allow that two phrases might signify different things, yet denote the very same thing. This indicates very clearly that there are contradicting interpretations of the same phrases. One interpretation says that they are different, the other that they are the same. Yet you allow that the contradicting interpretations are both correct.

It seems clear that you are using a different definition of "object" than the one rigorously employed within the discipline of semeiotic. Again, anything that is denoted by a sign--real or fictional, existent or imaginary--is an object in that technical sense. — aletheist

Why then did you insist on a distinction between "signification" and "denotation" in the other thread, when here you want any signification to be a denotation?

The only signs that theoretically could signify something without denoting anything are pure icons, unembodied qualities that would only convey themselves as they are in themselves. Any sign that stands for something else denotes that other object. — aletheist

So, you insist on a distinction between signification and denotation, then it turns out that there is no such thing as signification in common usage. All instances of signification are assumed to be denotations of objects. What's the point?

If this were true, then the author could not create those "images of characters" in the first place, and we could not think or talk or write about them afterwards. Again, the sign "Hamlet" denotes the fictional character in Shakespeare's play as its object. — aletheist

Let's start with a clean slate then. There is no such thing as signification. Words do not have meaning, they denote objects. is that what you want?

Otherwise it's pointless for me to say that an expression has meaning (signification) and you just overrule and say no, that's not meaning, it's the denotation of an object. We will never get anywhere like that. What is your rigorous definition of "object" which allows you to claim that any instance of meaning is a denotation of an object? If you and I both read the same expression, and I interpret it as meaningful, without denoting any object, and you interpret it as denoting an object, what is your rigorous definition which makes you right, and me wrong, if you are not just assuming that anything with meaning denotes an object?

(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.

(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do. — TonesInDeepFreeze

I told you already, extensionality provides a false premise. False premises produce unsound conclusions, which do not prove anything. When a human being judges two distinct things as having the same properties, and says therefore that they are equals, this does not make them into the same thing. The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing.

Ordinary mathematics regards '2+1' and '3' as having the same denotation, because we prove

2+1 = 3

In general, for any terms T and S, we infer

T = S

when we prove it and then we may say that T and S have the same denotation. — TonesInDeepFreeze

Clearly you, (and extensionality in general) have this backward. If we start with the law of identity, "a thing is the same as itself", as a fundamental premise, and we compare this with equality, which is a property that we assign to things, you ought to see this. When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness. We may not be able to account for all the potential differences between them, and thereby over look some, making a faulty judgement of "same".

So, we can truthfully say that a thing will be judged to be equal to itself, but we cannot truthfully say that things which are judged to be equal are the same thing. Therefore, when you say that T=S, you say that T has the same value as S within that system of judgement, and this means that the symbols have the same meaning within that system, but it does not mean that they denote the same object.

Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms. — TonesInDeepFreeze

This precisely, is the false premise. Being equal is a human judgement, and being equal does not imply being the same. We can proceed the other way, and say that being the same implies being equal, but we cannot proceed from being equal to an implication of being the same. This is because two distinct things can be judged as equal, when they are not the same. Therefore proving that two things are equal does not imply that they are the same (share all properties). It only implies that they share the properties by which they are judged to be equal. And there is your false premise.

This is a basic fact of the way that we use signs and symbols. We use the symbol "2" here, and we use it later in some other application. These are two distinct instances of that symbol, they are not the same thing. However, they have an equality in what they signify. Each distinct instance of using that similar symbol signifies "the same value". This means that the two instances have the same meaning. It does not mean that they denote the same object.

So, proving that two distinct yet similar instances of a symbol "2", have an equal value, only proves that they have the same meaning. It does not prove that they denote the same object.

Of course in natural language and everyday discourse there may be vagaries that make definitive determinations difficult or impossible. So if you want to demand a context in which there are no vagaries, then of course all bets are off with natural language usage. So to proceed with understanding certain principles, of course we must assume, for sake of discussion, some context in which we are not thwarted by such vagaries as you mention. — TonesInDeepFreeze

This is counterproductive. If in reality, language use is filled with vagaries, and we want to discuss the truth about language use, then we need to account for the reality of those vagaries. To assume a context without vagaries as your prerequisite premise for proceeding toward an understanding of certain principles of language use, is simply to assume a false premise. Therefore by adopting such a position we proceed toward a misunderstanding rather than an understanding.

Similarly, of course I take it for granted that we already understand that 2+1 = 3, either by proof or by common mathematical knowledge. It is from that understanding that we then observe that '2+1' and '3' denote the same number. — TonesInDeepFreeze

In the case of Henry Fonda, we have observed with our senses, the very object being referred to. In the case of numbers we have not observed any such objects. You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. Obviously, there are no such objects, the numerals have meaning dependent on the context of usage, just like any other symbols. They do not denote any objects, and your so-called understanding is actually a misunderstanding.

But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number. — TonesInDeepFreeze

Claiming a denotation when there is only meaning, is a false premise.

The claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person is not an argument! It is a conclusion. It is a conclusion from the premise (however it has been established) that Henry Fonda is the father of Peter Fonda. No one every suggested otherwise! — TonesInDeepFreeze

Do you understand the fallacy of "begging the question", assuming the conclusion?

I am treating "2+1" and "3" as signs here, and I already acknowledged that their signification is different. At issue is whether their denotation is different. What "+" represents in isolation is irrelevant, all that matters here is that I can point to the same group of items and truthfully say both "that is 2+1 apples" and "that is 3 apples." — aletheist

To be clear, what is at issue is whether there is a denotation at all (when denote is defined as you do). Read the above.

The "+" is not irrelevant, it must be accounted for in your interpretation. You cannot simply leave words out of a phrase, in your interpretation, to make it say what you want it to say, or denote what you want it to. I really do not see any logic to your claim that two expressions can have distinct significations, yet denote the same object. I can see how "I did X", and "I did Y", both refer to the same object with "I", but each signify something different. Since each expression signifies something completely different, if we replace what is signified with "denoting an object" as you seem inclined to, then we do not come up with the same object. How do you come up with this idea that two phrases which signify something completely different actually denote the same object. I would call that contradiction.

As I stated clearly in the last post. A group of three apples does not truthfully represent "2+1". If that's not obvious to you, go back to grade school and find out how they represent "2+1".

Indeed, but as the Fonda example has brought to light, Metaphysician Undercover apparently confuses denotation and signification. The result is wrongly denying that two different expressions signifying different interpretants can nevertheless denote the same object. — aletheist

Actually it has become very clear now, that you and Tones are the ones confusing denotation and signification. Clearly, in our use of mathematics there is signification without denotation. You and Tones are seeing an object denoted by "2+1", when there is none. That is misinterpretation.

However, we can point at a collection of three apples and say both "that is 2+1 apples" and "that is 3 apples." — aletheist

That's not true, because the operation signified by "+" is not evident in the group of three apples, so it is not a true representation of "2+1". It is just a representation of "3". If you were teaching children you would not show them a group of three apples and tell them this is 2+1.

Moreover, we can substitute "2+1" for "3" in any proposition without changing its truth value or in any equation without changing its result. What should we conclude from this? — aletheist

We might say that expressions which signify equal value can be substituted, within that value system . We cannot conclude that because the expressions can be substituted, they signify the same thing. They are only "the same" in relation to that value assigned to them.

If I need assistance, and Tom, Dick, or Harry, will do, each having equal value for the task, I ask for Tom, Dick, or Harry, as they are interchangeable in relation to this value. They each make "the task will be done" true. This does not mean that each of them is the same thing as each other.

Simple--in semeiotic, anything that is denoted by a sign is, by definition, its object. Since all thought is in signs, anything that we can think about--real or fictional, existent or imaginary--is an object in this sense. — aletheist

To adhere to the distinction you made for me in the other thread, in much usage of signs, probably the majority actually, the signs have significance without denoting anything. For instance in "I'm going for a walk", the only object denoted is "I". And in your example of fictional writing, there are no objects denoted. The author simply builds up images of characters without denoting any objects.

Then the example is irrelevant to the issue we are discussing, that "2+1" denotes the same object as "3".

You joined the discussion a bit late, and seem to be missing the issue.

Again, in semeiotic a subject is a term within a proposition that denotes one of its objects. — aletheist

Sure looks like fancified Platonism to me, if a subject must denote an object.

No one is claiming that fictional, imaginary things are real. In fact, being fictional is precisely the opposite of being real. That which is fictional is such as it is only because someone thinks about it that way; Hamlet was the prince of Denmark only because Shakespeare created a story in which that was the case. By contrast, that which is real is such as it is regardless of what anyone thinks about it; Platonism is one form of mathematical realism in this sense, but not the only one. — aletheist

I really don't understand your position. You assume that fictional characters are objects, but you deny that they are real, and you deny that they are existent. How do you validate your claim that they are objects?

Again, what anyone knows or does not know is beside the point. Since it is fact that Henry Fonda is the father of Peter Fonda, by definition (in semeiotic) the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object. — aletheist

These are nonsensical assertions. You are asserting that it is a fact that these words refer to these objects regardless of how people use the words. The issue is whether or not "the father of Peter Fonda" and "Henry Fonda" necessarily represent the same object. That you can define them as representing the same object, and insisting that this is a fact, is irrelevant I don't see any point in discussing the soundness of a logical argument with someone like who, who simply insists that the conclusion is a fact, and that's all there is to it.

If you want to start with the premise, that Henry Fonda is the father of Peter, we can do that. But then the example is irrelevant to the question of whether "2+1" denotes the same thing as "3", because we are not starting with that premise. We are arguing whether or not this claim is true, or logically sound. So we cannot start with the premise that "2+1" denotes the same thing as "3" because that would be a fallacy of begging the question.

So, to make the example relevant, we must start with the two expressions, "father of Peter Fonda", and "Henry Fonda", and you need to demonstrate how they necessarily refer to the same object, without begging the question. Insisting that it is a fact is simply begging the question, and that is a logical fallacy. So your procedure up until now has been completely useless.

I never said anything about persons or equality. I merely made the point--which is utterly uncontroversial (in semeiotic)--that since Henry Fonda is the father of Peter Fonda, the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object, regardless of whether someone else knows it. — aletheist

Yes, yes, keep begging the question, it really doesn't bother me if you do. You're only fooling yourself.

That latter is a bit disingenuous. If I say Socrates is a Greek philosopher, someone might object because they think I might have meant Socrates the cat philosopher. That's not really a good objection, if you fully qualified everything there would be no end to it. — fishfry

Let me give you a more relevant example. Let's consider an experiment in quantum physics. Consider that a photon is emitted by an emitter, and a photon is absorbed by a detecting machine. Each instance involves an equivalent amount of energy, so the assumption is that the two are the same photon. Then comes the difficult task of determining the continuous existence of that photon between point A and point B, which is produced by the idea that they are the same photon. But there is no need to assume that the two are the same photon, likewise there is no need to assume a continuous existence of the photon between point A and point B. It is only this (what I call odd) way of looking at things, that if there is a quantifiable value here, then an equal value over there, these must represent the same thing, which promotes the idea of the continuous existence of a photon between these two point.

Henry Fonda IS the father of Peter and that's that. — fishfry

As explained above, to premise that "Henry Fonda IS the father of Peter" is no different from premising that "2+1 IS 3". But since what I am looking for is an indication that 2+1 really is the same thing as 3, some sort of logical argument, that's simply begging the question. So the issue is to demonstrate logically, how it is that the two distinct expressions "the father of Peter Fonda", and "Henry Fonda" both refer to the exact same thing, and how this is relevant to the case of "2+1" and "3".

No, it does not. Hamlet, the fictional character in Shakespeare's play, is the object of the sign that is the first word of this sentence. — aletheist

Where's your grammar? Fictional characters are known as subjects, not objects. Your claim that "Hamlet" refers to an object is unsupported by any conventional grammar.

Even if I grant you that fictional, imaginary things may be called objects, my point was that some form of Platonism, as an ontology is required to support the claimed reality of such objects. So this line of argument is not really getting you anywhere.

At the risk of belaboring the point, it is an all-too-common nominalist mistake to insist that if abstract objects are real, then they must also exist. These are two very different concepts--whatever is real is such as it is regardless of what anyone thinks about it, while whatever exists reacts with other like things in the environment. Again, there are varieties of mathematical realism other than Platonism. — aletheist

Trying to establish a separation between "real" and "existent" just muddies the water by creating ambiguity, and is counterproductive toward understanding. As well as being "real", ideas, concepts and abstractions are obviously "existent". They have a significant effect on the physical world as clearly demonstrated by engineering.

So defining "existent" as having causal interaction, then attempting to remove ideas from this category is a mistake because ideas obviously have causal interaction. Then this proposed separation between "real" and "existent", which would put ideas into some category of eternal inert objects which cannot have any influence in our world in any way, is just child's play. It's an imaginary scenario which in no way represents reality.

I agree that they signify different interpretants, but this does not preclude them from denoting the same object. — aletheist

As I said, they may denote the same object, but we do not have the premises required to conclude that they do. In logic we cannot assume other premises which are not stated. We have a person denoted as "the father of Peter Fonda" and we have a person denoted as "Henry Fonda". We have no other information. So the conclusion that they both denote the same thing is extremely unsound, because it is not derived from valid logic. It is invalid.

t is a fact that Henry Fonda is the father of Peter Fonda, so by definition, it is also a fact that the signs "Henry Fonda" and "the father of Peter Fonda" both denote the same object. — aletheist

In logic, assertions do nothing for you. They are proposals, propositions which must be judged for truth or falsity. TonesinDeepFreeze has been asserting that "2+1" denotes the same object as "3" does, in a similar way. They very clearly each signify something different. The only attempt by Tones, to support this conclusion with a premise, was a vague reference to extensionality. But a premise which states that two equal things are the same thing is clearly false, making that argument unsound, by having such a falsity as a premise. To say that the person denoted as father of Peter Fonda, and the person denoted as Henry Fonda, are equal, as human beings, does not justify the claim that they are the same person.

If you just keep asserting as a proposition "it is a fact that...", and you expect me to take that proposition as a premise for an argument, then you're wrong. I will not. You need to demonstrate the truth of it. Adding the emphasis "it is a fact" does nothing for your case. I am very certain that two things with the same value are not necessarily the same thing, as I can give you endless examples, so that is a false premise.

But here I find myself inclined to see his side of it. — fishfry

Fishfry! Never in a hundred years did I think I'd see this day. Let's go, I'll buy you a beer.

I might know who Henry Fonda is, but I might not know he's Peter Fonda's father. I can see Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda." — fishfry

The point is that "the father of Peter Fonda" gives different information from "Henry Fonda". The latter gives nothing, just the name of a person. The first expression also denotes a person, as well as the second expression denotes a person. But the information required to conclude that they are one and the same person is not provided. Even if we add the further premise, "Henry has a son Peter", the condition of reversibility, equality, is fulfilled, but we still cannot conclude that they denote the same person. There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound.

." Again, "Henry Fonda" and "the father of Peter Fonda" denote the same object, even though what they signify about that object is different. — aletheist

Again, your argument that they denote the same object is fallacious. They may or may not denote the same object. They clearly signify something different, and we do not have the premises required to conclude that they denote the same object. Therefore your conclusion that they denote the same object is fallacious.

This is false, since it is not necessary for something to exist--in the metaphysical sense of reacting with other like things in the environment--in order to be the object of a sign. It does not even have to be real--it could instead be fictional, as some philosophers consider mathematical objects to be. — aletheist

This only supports my point. To justify calling an imaginary thing "an object" requires some form of Platonism.

The view that numbers are real, independently of any mental activity on a human's part, is what is generally known as mathematical platonism. The point is, this is unpopular in today's academy; there are many very influential mathematicians, who are far greater experts than I could ever hope to be, who are intent on showing that it's mistaken. But according to this article Benecareff's influential argument against platonism was made 'on the grounds that an adequate account of truth in mathematics implies the existence of abstract mathematical objects, but that such objects are epistemologically inaccessible because they are causally inert and beyond the reach of sense perception.' In other words, this argument denies that we can have the innate grasp of mathematical truths that Frege asserts in the paper mentioned above. That's the 'meta-argument' I'm trying to get my head around. — Wayfarer

That mathematical Platonism is unpopular in today's academy presents an odd dilemma for mathematicians. Platonist principles support a huge part of modern mathematical systems, underpinning extensionality and set theory, to begin with. These mathematical axioms require that a term signifies an object. Only Platonism can support this prerequisite. So, if there are proficient and influential mathematicians who openly deny Platonism, then these same mathematicians must be prepared to revisit, denounce and replace, all the fundamental mathematical axioms which are based in Platonism, or else they are simply being hypocritical.

Completely wrong, denotation and signification are two different aspects of a sign, corresponding respectively to its object and its interpretant. This is Semeiotic 101. — aletheist

That's bullshit 101. In logic, there is no object, we have subjects. To denote is simply to be a sign of.

I offered no argument at all, I simply stated a definition--if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object. This is also Semeiotic 101. — aletheist

Then you're not addressing the issue we've been discussing. We've been arguing the truth or falsity of of a very similar principle. The "Fonda" example was provided as an argument for the truth of it. As I've shown, it's a fallacious argument.

We were arguing the truth or falsity of the principle of substitution, which is the basis of extensionality. It is claimed that if two signifiers signify things of equal value, they are exchangeable, therefore they signify the very same object. It is very clear to me that this is a false principle because "equal" is assigned according to some system of judgement, so only the properties deemed significant within that system are accounted for, and this is insufficient for the conclusion of "the very same object". I find it utterly amazing, and rather distressing, the number of people in this forum who cannot apprehend this simple fact.

Now you are arguing a slightly different form of that principle. "Truth value" is something judged. Propositions are stated. Predications are of subjects. So within a logical system "truth value" concerns what we say about subjects, not objects. Unless absolutely every property of a given object is stated (a task humanly impossible), so that an infallible judgment can be made, your proposed principle: "if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object" Is clearly unacceptable as false. You have not provided the means for closing the subject/object gap.

Again, this confuses denotation with signification. — aletheist

Denotation is a form of signification.

In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. — aletheist

As I said, this is only the case if there is a premise which states that Henry Fonda is the father of Peter Fonda. But that is begging the question, which is respected as a fallacy.

Therefore the argument that "the father of Peter Fonda" denotes the same thing as "Henry Fonda" is a fallacious argument, by means of begging the question. The argument relies on assuming the conclusion.

As I explained, they do not denote the same object. One denotes the father of a person called Peter Fonda. The other denotes a person named Henry Fonda. That they denote the same object requires a further premise, that the father of Peter Fonda is the person named Henry Fonda.

Without that premise, the conclusion that they denote the same thing is invalid. And adding that premise is to beg the question. So the argument that they denote the same object is fallacious.

Are you going to address the points I made or not, Tones?

Do you apprehend the flaw in your example, and the difference between what "the father of Peter Fonda" denotes , and what "Henry Fonda" denotes?

In itself, yes; but we can still "divide" it at will to suit our purposes. — aletheist

To identify a point on a line is not to divide that line. So it's not really a matter of dividing which you are talking about.

For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space. — aletheist

The proposed units would be arbitrary, but I do not think you could call them discrete. And, according to the issues brought forward by special relativity, the supposed "same unit" would be different depending on the frame of reference, or more precisely, we could not determine the "same unit" from different frames of reference. This makes the whole idea of measuring the continuum through the means of units rather difficult. I suggest that if the application of units works well for measuring space and time, they are probably not actually continuous.

The speculation which is the reverse of yours is that continuity is what is artificial. The continuum is something created by human minds, and physical existence contains no such continuity. Problems such as Zeno's paradoxes arise because we apply principles of continuity to a physical world which is discontinuous.

No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude. — aletheist

If this is true, what you describe here, then it is impossible that "a true continuum is infinitely divisible. If marking points on a continuous line does not constitute dividing it, then there is nothing to indicate that the continuous line is divisible at all. And if dividing it once would break it's continuity, then a continuum cannot be infinitely divisible because dividing it once would prove it to be discontinuous.

Therefore it is a contradiction to say "a true continuum is infinitely divisible". We ought to say instead, "if it were divisible it would not be a true continuum". A true continuum is indivisible.

It is infinitely divisible, but not actually divided. — aletheist

Again, this is contradiction. If it cannot actually be divided, then it is false to claim that it is divisible, in any sense.

These are designated the 'primary attributes' of objects, and distinguished, by both Galileo and Locke, from their 'secondary attributes', which are held to be in the mind of the observer.

...

And through the quantitative method of science, the ability to reduce an objective to its mathematical correlates, the certainty provided by logical prediction can be applied to phenomena of all kinds with mathematical certainty (which is, I think, the point of Kant's 'synthetic a priori). It's the universal applicability of these logical and mathematical procedures to practically any subject which opens access to domains of possibility which would be forever out of reach to a mind incapable of counting. — Wayfarer

If, the distinction between primary and secondary attributes is broken down in this way, so as to allow for the universality of mathematical applications, then why conclude that all is "of the object" rather than all is "of the mind". As javi2541997;511801 indicates, Berkeley demonstrated that all is "of the mind" is the more logical conclusion. The other conclusion, that all is "of the object", requires the unsubstantiated assumption made by Kant, of the thing-in-itself, noumena, an assumption rejected by skeptics.

I distinctly did NOT say that. And you put that misrepresentation in quotes to fabricate something I did not say. — TonesInDeepFreeze

I apologize then, I misunderstood. I thought you meant that "2+1" could be interpreted as eithe of the following, (1) or (2).

For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.) — TonesInDeepFreeze

Now I realize you are insisting that it is neither.

'The father of Peter Fonda' denotes the value of the function (call it 'the father of function') applied to the argument Peter Fonda. That value is Henry Fonda. — TonesInDeepFreeze

This is clearly incorrect. "The father of Peter Fonda" denotes that there is a person who has the position, the special relationship of being the father of the mentioned person, and this person who is the father of the mentioned person is your subject. It does not say that this person is Henry Fonda, so you cannot jump to that conclusion. If you knew someone named Henry Fonda, it would be a logical fallacy to jump to the conclusion that this man is the referred subject. You have not made the required logical connection, to determine that your subject is the same person as the one you know as Henry Fonda.

For about the seventh time now: '2+1' denotes the value of the function.. — TonesInDeepFreeze

This is false as well, and you just don't seem to get it. Take a look at your example of "the father of Peter Fonda". The thing which you claim as "the value", is clearly not signified, because the premise required to produce the logical conclusion is not stated in the argument. We need a further "unstated" (that's the way I use quotations, to signify special significance) premise to make your assertion a valid conclusion. In your example, the required premise might be "the person you know as Henry Fonda is the father of Peter Fonda". Then you can validly conclude that when some one says "the father of Peter Fonda", this is the person you know as Henry Fonda.

The thing which you seem to have no respect for, is the fact that "the father of Peter Fonda" does not refer to "Henry Fonda". This is very clear from the fact that one stated premise in a logical argument cannot refer to a conclusion. "Socrates is a man" does not refer to the conclusion "Socrates is mortal". That is because the expression does not include everything required to make that reference. Nor does "2+1" refer to the value signified by "3", because it does not include everything required to make that reference.

When we jump to a logical conclusion without stating the required premises, error is possible. You know someone named "Henry Fonda"; you jump to the conclusion that this is the man referred to by "the father of Peter Fonda", and mistake is possible. Rigorous logic seeks to exclude the possibility of mistake, not to create the possibility of mistake. The principles you are arguing for create the possibility of mistake by removing the need for the statement of premises. If some premises can be taken for granted, and not stated, as you seem to believe, then those premises cannot be judged for truth of falsity, and error is possible.

But '2+1' is not a description of a procedure. — TonesInDeepFreeze

Well, clearly "2+1" does not refer to a value. That is an invalid conclusion as I explained above. So, if it does not refer to a procedure, as I think it does, is it possible that we can find a compromise?

The term itself doesn't denote that it has a result. — TonesInDeepFreeze

OK, so what in the expression "2+1" denotes that there is a result. Grand Minnow was insisting that the expression denotes a result. I don't see it in the signification. Now you're pretending to be someone else, so that your inconsistency is not so glaring. Grand Minnow can argue that a result is signified and Deep Freeze can argue that an operation is signified. How's that?

The usage "result of an operation" is an informal way of referring to the value of the function for the arguments. — TonesInDeepFreeze

A function is a process. Grand Minnow kept insisting that "2+1" does not signify a process. That's why I say there is inconsistency. But clearly an "operation" or "function" is a process, and that's what is signified with "+".

The "value" of the function is not signified, because it must be figured out by carrying out the operation which is signified. If I say add some sugar to water, and bring it to a boil, the value (result) of that operation is syrup. But I'm not telling you "syrup", I'm telling you the procedure to make it. To obtain that value, syrup, you must carry out the operation referred to first.

For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.)

So I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm). — TonesInDeepFreeze

OK, I'm fine with "operation", so long as you recognize that what is signified is a a procedure, or operation, and as you say, this is "a procedure for determining the value of a function applied to an argument". The value is not signified, the "procedure for determining the value" is what is signified. Do you agree?

Do you even know what the use-mention distinction is? I — TonesInDeepFreeze

Of course I do. In philosophy we use a different convention. I use " " to signify a concept rather than a physical thing. I'm trying to conform to your convention but I'm a bit sloppy and missed one. Call it a typo.

There are an infinite number of ways to refer to the number 3. That doesn't mean they don't refer! Your argument is so daft! — TonesInDeepFreeze

Here's your inconsistency. You distinctly said "2+1" refers to "a procedure for determining the value of a function applied to an argument."

Now it's my turn to ask you, do you understand the difference between a procedure (function, or operation), and an object? Aristotle demonstrated a fundamental incompatibility between these two. A procedure cannot be an object, and an object cannot be a procedure because of this fundamental incompatibility. If "3" refers to the number three, and this is an object, then the procedure for determining a value, referred to with "2+1", cannot be the same thing as what is referred to with "3".

No he doesn't. If he does, he's wasting precious billable seconds. Instead, he just goes ahead to add the numbers. — TonesInDeepFreeze

Have you ever seen a ledger? Every account must be stated and balanced. Call it redundancy if you want, but there must be no room for error.

Are you serious? Are you trolling? — TonesInDeepFreeze

Of course I'm serious. You just told me there is an infinite number of ways to say "2+1", and I assume an infinite number of ways to say "3+1", so I ask you what distinguishes one from the other? Why is "3+1" not just another one of the infinite ways of saying "2+1"? That you do not answer means that you do not know.

No, the term '500+ 894+202' already denotes 1596. — TonesInDeepFreeze

Here is your inconsistency. Above, you said things like "500+ 894+202" denote "a procedure for determining the value of a function applied to an argument", which I accept. Now you are claiming that it actually signifies the value. What you say now is clearly false, because the procedure must be carried out before that value is derived.

It's just that the accountant doesn't know that until he performs the addition. The term doesn't start denoting only upon the knowledge of the account. The term doesn't spring into denotation every time some human being or computer somewhere in the world does a calculation. — TonesInDeepFreeze

What the person knows, is that "500+ 894+202" signifies the operation required to determine the value. Your claim that "500+ 894+202" represents the value is nothing but a misrepresentation. And, if you proceed in a philosophical argument with that misrepresentation of what "500+ 894+202" is known to signify, you are guilty of equivocation.

I left it out because it is a nonrestrictive clause. Further, any necessary termination is for a reason external to the process itself, usually to make an approximation. — tim wood

I don't know what you're talking about tim. To say "X will necessarily be terminated" seems very restrictive to me. Obviously, the cause of termination of the process is external, that's Newton's first law. But how's that relevant?

Ok. "Is said" seems gratuitous. π, I'm told, in decimal expansion never ends. To use it as a number, it's usually truncated at some point. That is just a number, nothing infinite about it at all, potential or otherwise. But why confuse the two? One stands in for the other to get an approximation. What is the issue about "potential" anything? — tim wood

This was Ryan's term. You'd have to go back to see what Ryan was talking about. Essentially Ryan suggested replacing infinities with infinite processes. Since the supposed infinite processes could never be completed they are assumed to be potentially infinite. I argued that every supposed potentially infinite process will for some reason or another, at some point be terminated. If this is the case then it is incorrect to even call them potentially infinite.

"But a process which is said to be potentially infinite, cannot truthfully be said to be potentially infinite." Eh? Sure it can. Or do you mean that the never ending decimal expression of π actually ends? — tim wood

You left out the important phrase: "which will necessarily be terminated". The infinite process would continue forever, by definition. Since forever never arrives Ryan says we ought to call it potentially infinite. Ryan suggested that we could put an end to a potentially infinite process, by rounding of pi for example, yet still say that it is potentially infinite. Obviously though, if someone puts an end to a process, it is not potentially infinite. So Ryan proceeded to distinguish between the rule which produces the supposed potentially infinite process, and the process itself, trying to place the potential for infinity within the rule rather than the process.

I'm offering you help here, though I doubt you'll take it in. — GrandMinnow

If it makes sense, I'll take it. But so far all you've offered is inconsistency. Let me see if I can follow you.

'2+1' denotes the result of the operation — GrandMinnow

I see the number 2 denoted, and the operation + denoted, and the number 1 denoted. So there is clearly an operation denoted. What denotes that the operation has a result?

Would you agree that a finite operation is distinct from an infinite operation, the one having a result, the other not? If this is the case then there is a need to distinguish between an operation with a result, and one without a result.

You got it exactly backwards. Our method does not lead to '2+1' denoting infinitely many things. '2+1' denotes exactly one thing. On the other hand, 2+1 is denoted infinitely many ways:

2+1 is denoted by '2+1'

2+1 is denoted by '3'

2+1 is denoted 'sqrt(9)'

2+1 is denoted by '((100-40)/3)-17'

etc. — GrandMinnow

It appears to me, like 2+1 demotes exactly nothing then. You can say the same thing in an infinite number of different ways, but none of these ways refer to anything. Each expression simply say I am the same as the infinity of others. If one of them refers to anything real, then they all must refer to something real, and you have an infinity of equivocation, with an infinity of different things referred to by on signification. Even the numeral "3" must refer to the result of an operation, exactly as the others, so there is nothing to validate any object

So what makes 2+1 different from 3+1 then? Each can be said in an infinity of different ways, and there is nothing which is actually being referred to be either one. How can they differ?

If we want to know how much a company did in sales, the accountant starts by seeing that the company got 500 dollars from Acme Corp., and 894 dollars from Babco Corp, and 202 dollars from Champco Corp. Then the accountant reports:

500+894+202 = 1596 — GrandMinnow

This is incorrect. The accountant writes out 500+ 894+202=?, or x, or some other placeholder for the unknown, because the sum is unknown. But if it were like you say, that "500+ 894+202" already says 1596, then the accountant would not have to sum up the numbers, because the result of the operation would already be stated.

That's why your way of looking at things, if it were true, would render the equation completely unnecessary and redundant. By the time the left side was stated, (500+ 894+202) the right side would already be known, because you claim that the left side states the result of an operation. Clearly this is false, because equations contain unknowns, and this is how we solve problems, by carrying out the operations required to determine the unknowns. Obviously you are spinning a web of deceit.

One wouldn't honestly claim to know that the equation is true until one worked it out that it is true. Or to find a right side without '+' in it, then first one might have to perform the addition on the left side. This doesn't vitiate anything I've said. — GrandMinnow

How do you apprehend a need to work things out? If "2+1" says sqrt(9), how is there any need to work out any equivalencies?

But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.
— Metaphysician Undercover

Care to edit this? I do not understand the last part. — tim wood

Sorry I left off the suffix, 'ly'. Try this:

But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potentially infinite.

That better?

It was synthesis who raised the idea that God is an absolute. I was just saying that this idea, of God as an absolute was in synthesis' mind.

Perhaps you can explain yourself here. — synthesis

The maple trees look like they're dormant, but if you tap them they'll give you sap, so they're really not dormant. Looks are deceiving.

The Relative and The Absolute stand opposed to each other as that which we use intellectually (the Relative) and that which exist outside of our intellect (The Absolute). All things knowable (intellectual) are relative. These things that exist intellectually are constantly changing, exist in time, therefore their relative nature. — synthesis

I think you've got this backward. The absolutes are ideals, they are within your mind. Things outside your mind are relative. "All things knowable (intellectual) are relative." is an absolute which your mind has for some reason produced. "God" is an absolute which human minds have for some reason produced.

Accessing The Absolute is the goal of all spirituality and religion, as this is where the The Truth lies. And although you can never know this Truth, you can be with and part of it, a need that has apparently driven man's behavior for thousands of years. — synthesis

Once you see that the absolutes are within you, you'll have no problem to access them, just direct your attention that way, and learn how to ignore the external distractions.

Instead of bothering you guys, I think I'll go outside and consult with the Oak and maybe the Maple, as well. — synthesis

It's a very good time to consult with the maples, they've got much to offer. I'm about to go make some syrup myself. The oaks appear to be dormant right now so maybe they've got nothing to offer. Come to think of it, the maples appear to be dormant too. Looks are deceiving because you judge your sensations relative to your intentions. And your intentions may be misguided.

But I would say that while you can't take THE OBSERVATION any further, you can improve your intuitions. — Acyutananda

What do you think improving one's intuitions would consist of? Aristotle placed intuition as the highest form of knowledge in his Nichomachean Ethics. He looked briefly at the question of whether intuition is innate or whether it is learned, and decided it was a combination of both.

In western society we generally consider intuition to be instinctual. It is the inherited aspect of knowledge. When you say intuition grasps the truth of "2+2=4", this would mean that we instinctually accept this as true. However, we still need to learn the meaning of the equation. We are taught it in school, so the instinctual aspect is the attitude that we have toward learning. We accept the teacher (authoritative figure) as the authority, we have a desire to learn, we see the usefulness in what is being taught, so it appeals to our intuitions.

How do you propose that it is possible to improve one's intuitions? Would this be a matter of moral training, to improve one's attitude toward authority? Or what do you think?

I will tell him about your claim that infinities play no role in programs and see what HE has to say about that. — Gregory

You seem to have missed the gist of the conversation Gregory. I said that when they are rounded off, or "terminated" in Ryan's words, such as the example in the op, or using pi as 3.14, then the so-called infinities are very useful. But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.

2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number. — GrandMinnow

Ok, we we have the numbers 2 and 1 denoted, and the operation + is denoted. Where is the result of the operation denoted? It seems to me like you're jumping the gun. Jumping to the conclusion, assuming that the some result of the operation, 3, is already denoted when clearly it is not denoted

That's the reason why we need to denote = 3, if we want to denote some result, because "2+1" on its own does not say 3. Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". Therefore "2+1" would denote an infinite number of things, and that would make interpretation impossible. Furthermore, equations would be absolutely useless because the right side would just be saying the exact same thing as the left side, along with all the infinite other things that are equal. What would be the point to an equation in which the right side represented the exact same thing as the left? You'd never solve any problems that way, because the problem would be solved prior to making the equation. If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign.

-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it. — Ryan O'Connor

But spitting out numbers is not something useful. Useful is the application of the numbers towards counting or measuring, or something like that. If the computer is tasked with counting something and does not complete the task it hasn't been useful.

-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite) — Ryan O'Connor

But what good is that?

n the end, I think you're splitting hairs here. What's your point? — Ryan O'Connor

I can't even remember now, but I believe I said it would be good to rid the system of infinities and you said there is no problem with working with infinities so long as we recognize that they are merely potential.

But what's the point to working with infinities? If an infinity represents an uncompleted tasked, then isn't it better to complete the task before proceeding. After a while the unfinished tasks start to pile up and become a little overwhelming. And if it is a task which is impossible to complete, then to give oneself an infinite task is to set oneself up for failure, so we ought to address the conditions by which this happens so that we can avoid it.

π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. — Ryan O'Connor

Obviously, that's not a real solution.

Also, who said math had to be practical? — Ryan O'Connor

This is probably the crux. "Math does not have to be practical". There's a fundamental element of free choice which lies at the base of all of our understanding of everything. "Has to be" is thereby excluded. And so, we do not have to do anything, nor do we have to figure anything out, or anything like that. One can refuse to move and die if one wants. However, we choose to try and figure things out, we choose to try and understand the nature of reality, and mathematics plays a very big role here. So we need to choose the appropriate mathematics.

Of the mathematicians, the people who dream up axioms, and produce elaborate systems, some might have the attitude that math does not have to be practical, and others might have the attitude that math ought to be practical. But the idea that mathematics does not have to be practical is just an illusion. Each such mathematician will choose a problem, or problems to work on, as that's what mathematics is, working on problems. And problems only exist in relation to practice, as that's what a problem is, a doubtful aspect of practice which needs to be resolved. Without the influence of practice, the need to resolve the issue does not arise, therefore there is no problem. So the reality of the situation is that since mathematicians work on resolving problems, and problems only exist in relation to practice, math is always fundamentally practical, and this fact cannot be avoided. That's why math is classified as an art rather than a science. Despite the huge amount of theory which goes into it, it is theory which is always directed toward solving problems. Therefore, despite the fact that math doesn't have to be practical, it always is practical. If the people who dreamed up axioms and other systems weren't doing something practical (resolving problems), they would have come up with something other than mathematics.

The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2. — GrandMinnow

In general, "2" denotes a number, and "1" denotes a number, but in this particular circumstance, "2" does not denote a number, and "1" does not denote a number. Therefore you equivocate.

A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted. — Ryan O'Connor

We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.

If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated. — Ryan O'Connor

I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.

But you don't know anything about the formulation of classical mathematics.

...

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation. — GrandMinnow

As I said, if this point is of relevance then the discussion is pointless.

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction. — GrandMinnow

I can't believe that you do not understand the contradiction. Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. Do you understand this?

A process is a sequence of steps. — GrandMinnow

This is false. A process may be described as a sequence of steps. The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. That's why different people can use different methods to resolve the same mathematical equation.

Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc. — GrandMinnow

I don't see any need to consider an abstraction as an object. Abstraction is simply how we interpret things, and there is no need to assume objects of meaning as a fundamental part of the interpretive process.

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing. — GrandMinnow

You agreed that they are different things which have the same result, or the same value. If they are different things, then having the same result, or the same value does not justify calling them the same thing.

Here's what you said:

We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

Are you taking that back now? Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6, instead of what you already agreed, that they are distinct things with the same end result? You know the truth in this matter, why try to deny it?

Properties are not things that are physical objects. — GrandMinnow

Then why treat properties as if they are any sort of object? You treat numbers as if they are some sort of objects, when really they are a property of the thing which is numbered.

I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional. — GrandMinnow

I've argued elsewhere that the axiom of extensionality is a falsity. It is the means by which you say that two equal things are the same thing, which is obviously false. So it's not necessarily that I do not understand extensionality, but I apprehend it as based in false premises.

That is, the principle of "substitute equals for equals" holds. — GrandMinnow

In other words, equal things may be considered as the same thing. And that's clearly false.

If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields. — Ryan O'Connor

There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. Here's an example. The ancient Greeks used circles to model the movement of the planets, and Aristotle proposed that the orbits were eternal circular motions. It turned out that these models were wrong, therefore it was not true for them to have been saying that the orbits were circles even though this concept was employed and enabled prediction.

No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button). — Ryan O'Connor

Again, there is an issue of truth here. If the process is terminated then it is untrue to say that it is potentially infinite. And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. Therefore t is false to say that the potentially infinite process is useful.

But nonetheless banishing infinity from mathematics is a move of an ostrich — Gregory

No, the opposite is the case. Ignoring the fact that infinities in mathematics is a very real problem, is the type of ignorance which is analogous with the ostrich move.

But he/it doesn't, so the issue of passing particular points is no different from passing any point, and yet all those other points are never mentioned. Why is that, do you suppose? Achilleus - or the Arrow - seems to have no problem whatever passing those. Zeno's then, just an entanglement with words. — tim wood

What do you think "passing a point" means? Do you mean to say that there are physical points out there, which the arrow can be seen flying by? If so, then you ought to be able to show empirically, the physical existence of such points, and I don't think there will be an infinity of them. If these points are just imaginary, then the arrow doesn't really fly by them and you have created a false scenario, by describing the arrow as flying by points.

I propose that the truth is that the points are imaginary. If this is the case, then any method of measuring motion, velocity and such; which employs points, is really giving us a false measurement. We might be able to find real physical points, which if they exist, would validate such a method, but then these points would not be infinite, so that scenario with infinite would become irrelevant, because we'd have to make a new method of measuring velocity based on empirically verified points. As I explained to you earlier, this is pretty much what relativity theory does, but each empirically verified point turns out to be a different frame of reference, and that the points are at rest relative to each other is very doubtful due to the observed phenomenon of spatial expansion.

You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation. — GrandMinnow

I have no formulation, and no desire to present one. The op asks if something has been proved, therefore we are invited to be critical of formulations which claim to prove that. And there is no need to offer an alternative formulation to point out problems with an existing one.

Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you. — GrandMinnow

As I said, you equivocate:

I said explicitly that '1' and '2' do each refer to a distinct object. — GrandMinnow

2+1 is a number. — GrandMinnow

Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. But I've been trying to go easy on you and settle for the lesser charge of equivocation. If "1" and "2" each signify distinct numbers, then there are two distinct numbers represented by "2+1", so it is contradictory to say that "2+1" represents one number, because there are two numbers represented here.

It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2. — GrandMinnow

That the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object.

The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6. — GrandMinnow

Sure, the resulting value of each is 3, but that's not the issue. Your claim is that "=" signifies identical to. 6-3 equals 2+1, but what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. Therefore it should be very clear to you that "=" does not signify identical to.

If you say that they have the exact same value, then we are using "equal" in the way I suggested. You and I have the exact same value in the legal system, therefore, as human beings we are equal, just like 6-3 has the same value as 2+1 in the mathematical system, but in neither case are the two equal things identical.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

OK, let's go with this then. If you do something, and derive a result, this is necessarily a process. So you are very clearly talking about two distinct processes represented by "2+1", and "6-3". Two distinct and different processes can have the same end result, and so those processes can be said to be equal. Does this imply, that in mathematics you judge a process according to the end result? If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op?

Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there exists a God. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in human reasoning. — GrandMinnow

I don't see the difference. You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". Each of you will try to justify the claimed existence of your imaginary object. You are not showing the necessity required, which the theologians show, so you are not doing a very good job of it.

We prove from axioms that there is a unique object having a certain property, and we name it '6'. — GrandMinnow

This is so contradictory to what you've been arguing. You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. And it isn't just a matter of different names for the same thing, because "4" and "2" must each name a unique thing, so it's impossible that "4+2" is just a different name for "6". How can you now claim to be able to prove that there is a unique object named "6", when you've been arguing that all these different things are the same as 6, by virtue of equality. You are getting yourself so tangled up in a web of deceit, that's it's actually becoming ridiculous.

How does "sqrt" change from signifying an operation, to signifying an object, simply by qualifying (or quantifying, if you prefer) it with a (2), without equivocation?

It's the same sort of problem which GM has with 2+1. Each of the symbols "2", and "1" refer to a distinct object. But GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. How is this not equivocation?

We measure the car at 60mph and maybe that's accurate to within a small margin of error. — tim wood

I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction.

My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect? — Ryan O'Connor

I really don't get your question. I was talking about points, not particles, so your question has some underlying presumptions which I don't follow.

Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source — Ryan O'Connor

I addressed this already. The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. It is implied that there is an underlying substratum which validates this modeling, but the modeling itself, the quantum field theory, does not represent the underlying substratum, it represents the interaction of particles. Until we get accurate and precise modeling of the particles any speculation concerning the substratum is not well informed.

I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program). — Ryan O'Connor

OK, so we say that the rule calls for the computer to carry out an endless, or infinite process. We know that the computer cannot succeed in carrying out this request, because it will wear out first, so all the time spent will be wasted, for the computer to be trying to carry out a process it can't. So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished.

The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things, as you seem inclined toward, then the discussion is pointless. You'll keeping insisting that I am not seeing things the way that mathematics sees things therefore I am necessarily wrong. I've been exposed to enough of this already, and see no point to it.

(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values. — GrandMinnow

OK, I'll try to adhere to this formality. It is not the convention I am used to, but I'll try it.

2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. — GrandMinnow

No, "2" and "1" signify values. Or do they sometime signify values and other times signify arguments? If so how do we avoid equivocation? Anyway, I see no way that a function, which is a process, could have a value. That's like saying that + has a value.

You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. — GrandMinnow

Look, "2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. You cannot try to tell me that to take three away from six is the exact same thing as putting two together with one, or I'll tell you to go back to elementary school and learn fundamental arithmetic properly.. Your claim is clearly false, equals does not mean identical to, or the same as, in mathematics.

Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain. — GrandMinnow

I know that a function is a process. And I also know that the concept of process is incompatible with the concept of object. The two are distinct categories. Therefore it is fundamentally incorrect, by way of category mistake, to say that a function is an object.

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ... — GrandMinnow

I really don't know what you could possibly mean by this. The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven.

You are saying that the number of them is 6. — GrandMinnow

More correctly, the quantity is six. You assume that this quantity is an object, a number, which is something other than the quantity. But that's not the case, the quantity is six, the number is six, and both "quantity" and "number" mean the same thing here. Why assume that there is something other than a quantity, an object called 6? That makes no sense, where and how are we going to find this object?.

When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. — GrandMinnow

You're just making an imaginary thing, like God, and handing a property, "even " to that thing, like someone might say God is omniscient. When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. But the group clearly cannot be divided by three. If you say that "2" refers to some imaginary object, then you can assign to it whatever properties you like. You could make it infinitely divisible if you want. But if you're not adhering to any principles of reality, this is just useless nonsense. What good is assuming an imaginary object which you can attribute any properties to with total disregard for reality?

With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place predicate of equality, and indicates in the equation that the predicate of equality holds for the pair <2+1 3>. — GrandMinnow

Now you're doing the same thing again, you're claiming two nouns, 2 and 1, are one noun signified as "2+1". Clearly this cannot be the case without equivocation. Either 2 is a noun and it refers to an object, or it is not. But you can't have it sometimes being a noun, and sometimes not without equivocation.



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