The whole debate is exactly about the ontological status of abstracts. — Wayfarer

What more could be said about the ontological status of abstracts other than that they are real abstracts? Analogously what more could be said about real physicals other than that they are real physicals?

But if they’re real, then what kind of existence do they have? What does it mean to say abstract objects exist? — Wayfarer

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.

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 — aletheist

I agree with you, but thousands wouldn’t. I’ve had many exchanges with others on this Forum who believe that ‘existence’ and ‘reality’ are synonyms, and who can’t imagine what it would mean for them to differ.

What more could be said about the ontological status of abstracts other than that they are real abstracts? — Janus

Surely you can see how this poses a problem for naturalism? If you can’t see it, then sure, there’s nothing to discuss.

That is the point at issue! If numbers are real, but not corporeal, then it's a defeater for philosophical materialism - there are reals that are not material. — Wayfarer

This might help the case for some version of idealism (non-materialism).

They (numbers) don't exist in the same way that flowers or pens or chairs exist but are real nonetheless. — Wayfarer

This maybe the stumbling block for idealism (non-materialism).

Even if we were to all agree that immaterial numbers exist, we still have to contend with the fact that numbers aren't like "...flowers or pens or chairs..." We're not out of the woods yet.

:up: :ok:

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.

it is an all-too-common nominalist mistake to insist that if abstract objects are real, then they must also exist. — aletheist

... An all-too-rarely credited nominalist insight, rather.

Trying to establish a separation between "real" and "existent" just muddies the water by creating ambiguity, and is counterproductive toward understanding. — Metaphysician Undercover

ruining the good old word "exist". — Quine, On What There Is

Fictional characters are known as subjects, not objects. — Metaphysician Undercover

Again, in semeiotic a subject is a term within a proposition that denotes one of its objects. "Hamlet" is a sign, a subject of a proposition such as "Hamlet killed Claudius." The fictional character in Shakespeare's play is its object. Other subjects of that proposition are "killing" and "Claudius," which denote a relation and another fictional character in Shakespeare's play as their objects. The predicate is signified by the syntax, conveying that something called "Hamlet" stood in the relation of "killing" to something called "Claudius" within the universe of discourse, which in this case is Shakespeare's fictional play--not the real universe.

Your claim that "Hamlet" refers to an object is unsupported by any conventional grammar. — Metaphysician Undercover

It is firmly supported by what is known as speculative (theoretical) grammar within semeiotic, the science of all signs.

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. — Metaphysician Undercover

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.

Trying to establish a separation between "real" and "existent" just muddies the water by creating ambiguity, and is counterproductive toward understanding. — Metaphysician Undercover

On the contrary, I have found that carefully drawing the proper distinction between reality (whatever is such as it is regardless of what anyone thinks about it) and existence (whatever reacts with other like things in the environment) is extremely clarifying and helpful. Treating them as synonymous is what muddies the water by imposing nominalism, effectively begging the question against realism.

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. — Metaphysician Undercover

The first sentence is false, but the second is true. The kind of effects that ideas, concepts, and abstractions have on the physical world is obviously very different from the kind of effects that physical things have on the physical world. The former are not like things in the (physical) environment, so they do not exist in that sense. Nevertheless, some of them are real--they are such as they are regardless of what anyone thinks about them--but this does not require them to be "located" in a Platonic realm.

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

I don't think there is any problem with saying that numbers exist in logical space just as objects exist in 'physical' space. Numbers react with "like things" (other numbers) in their environment just as physical things do in theirs. Also number, quantity, is a significant, integral part of all physical interaction, so it's not like number (quantity, as opposed to the abstraction from quantity; numbers as distinct entities) doesn't exist in the physical world. As I said before there is an actual difference between three apples and four apples.

Surely you can see how this poses a problem for naturalism? If you can’t see it, then sure, there’s nothing to discuss. — Wayfarer

It depends on what you mean by "naturalism". For me number is an obvious part of nature, and abstraction is a natural part of being a self-reflective being. If you don't want to posit a supernatural reality, or you don't associate naturalism per se with the kind of reductive scientism that thinks everything can be explained in objectivist terms or even more extremely, in terms of physics, then what is the problem with naturalism? I see naturalism at its most general as simply a rejection of the supernatural as a separate or higher non-physical reality or realm.

The most striking thing about The Unreasonable Effectiveness of Mathematics in the Natural Sciences is that the argument is not about reason, so much as about Wigner's sentiment. That maths has been used with such success feels wrong to Wigner.

One is tempted to answer: so what?

On

Frege's 'laws of thought' — Wayfarer

A child who learned chess by watching adults play might insist that it is "self-evident" that the bishop stays on its original colour.

But it isn't. It's just how we play the game.

Frege might insist that there are primitive truths that are self-evident.

But there aren't. It's just how we play the game.

We might add a rule to Chess such that when the Bishop moves to a corner at the far end of the board it may move immediately to an adjacent square, changing colour. We could continue to play despite having broken the self-evident rule.

We add a rule to mathematics naming the square root of -1 'i'; and we continue to play the game.

What looks like an intuition — that the bishop stays on its own colour and that there is no root to -1 — is mere convention.

This topic lies parallel to my "1" does not refer to anything., where I advocated a constructivist approach after me ol' mate Witti. So I will transcribe something from there:

I'm picturing the issue as one of as direction of fit, as in Anscombe. That the number of things in the box is 3 is something we do; the direction of fit is from us to the world.

Add to that, that concepts are not things so much as a way of behaving; that is, concepts are best not considered as things in people's minds, but as ways of talking and acting.

Then we have a way of talking that goes "one, two, three" while pointing to each thing in turn. And we can use this way of talking to talk about lot of different things. And then we can talk about this way of talking when we find ourselves adding, then multiplying, then differentiating...

SO even though numbers are not things, we develop mathematics by treating them as if they are. And in the end they become things just by our having treated them as such.

So, the extension of "Wayfarer" is Wayfarer. The extension of "red" is each and every red thing. But the extension of "1"? Well it's literally every individual. And as such it seems to me, at least in my present mood, that the extension drops out of the game, and what we have is the intension, the rule, concept or game we play in counting.

And the consequence of that is that talk of extension in mathematics becomes fraught with ambiguity. Hence, Wittgenstein's argument that mathematical extensions must be finite, and hence his adoption of finitism, seems misguided.

...all this to say that Platonic Realism towards mathematical objects is not wrong, but just one way of treating those objects. It comes about by extending the notion of what is real to encompass numbers.

Add to that, that concepts are not things so much as a way of behaving; that is, concepts are best not considered as things in people's minds, but as ways of talking and acting. — Banno

Tosh. When Einstein discovered relativity, he was not ‘behaving’ or ‘acting’, or rather, his ‘behaviour’ and ‘action’ was mainly sitting and writing, and, I imagine, staring into space, and taking long walks. Many people ‘behave’ and ‘act’ like that, but that doesn’t give rise to a monumental scientific discovery. That was made by virtue of rational insight and unique mathematical intuition. Einstein had to devise a mathematical lexicon to articulate what he saw. And as is well known, those insights have proven predictive time after time. How many times have we read the headline ‘Einstein Proved Right Again’?

I think the issue around Platonic realism is simply that it torpedoes one of the beloved dogmas of empiricism, ‘no innate ideas’. There’s your ultimate sacred cow in our day and age. Read these snippets from the various sources I’ve referred to:

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)

And from the Internet Encyclopedia of Philosophy article on the Putnam-Quine Indispensability argument:

Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to debar any knowledge of mathematical objects. Thus, the philosopher of mathematics faces a dilemma: either abandon standard readings of mathematical claims or give up our best epistemic theories.

And are our ‘best epistemic theories’? Why, they’re grounded in neo-Darwinian materialism, which can never allow that we could have intuition of mathematical verities:

Mathematical objects are not the kinds of things that we can see or touch, or smell, taste or hear. If we can not learn about mathematical objects by using our senses, a serious worry arises about how we can justify our mathematical beliefs.

... Sets are abstract objects, lacking any spatio-temporal location. Their existence is not contingent on our existence. They lack causal efficacy. Our question, then, given that we lack sense experience of sets, is how we can justify our beliefs about sets and set theory.

Oh, what a quandary. So why not admit the obvious: that ‘our best empirical theories’ leave something fundamental out? Because then, ‘you’re opening the door to Religion’, and ‘empiricism might as well go out the window’. Well - splendid idea - better alternative than having to devise these tortuous rationales like the Indispensability Argument or Fictionalism.

It’s politics, at the end of the day. There’s certain ways you’re supposed to think, and if you don’t go along with that, woe betide unto you.

concepts are not things so much as a way of behaving; that is, concepts are best not considered as things in people's minds, but as ways of talking and acting.
— Banno

Tosh. — Wayfarer

Relativistic is a way of calculating relative velocities at speeds approaching c - a way of behaving.

I think the issue around Platonic realism is simply that it torpedoes one of the beloved dogmas of empiricism, ‘no innate ideas’. — Wayfarer

I think the issue around Platonic realism is simply that you wish it to torpedo one of your despised dogmas of empiricism, ‘no innate ideas’.

But this is not getting the conversation very far at all.

SO we might try this: if you think mathematical objects are real, perhaps we might look at how "real" works.

It's a real car, not a toy. It's a real dollar note, not a forgery. It's a real boy, not made of wood.

It's a real mathematical object, not...?

What?

*

Well, stop there if you must, but it looks like you are unable to respond to the criticisms I levelled:

1. Despite the title, Wigner is merely expressing a sentiment .
2. Edit - I left out Frege. What is self-evident to one person may be rejected by another. In particular Frege's desire to develop a logical basis for arithmetic and hence for maths as a whole has been show impossible. Maths cannot be derived in its entirety from self-evident truths.
3. There are alternatives — I offered by way of example good reasons to suppose that numbers do not refer to anything — and so no need to jump to your believe in Platonic Realities
4. The use of "real" in Platonic realism needs justification. It is not at all clear what it is that Platonic Realism claims is real...

1. Despite the title, Wigner is merely expressing a sentiment . — Banno

Bare assertion with no supporting argument.

I offered by way of example good reasons to suppose that numbers do not refer to anything — Banno

You simply referred to behaviourism, which has been obsolete since the 1940's.

The use of "real" in Platonic realism needs justification. It is not at all clear what it is that Platonic Realism claims is real... — Banno

It's not clear to you. The Indispensability Argument is a good starting point. But this controversy is about something.

Bare assertion with no supporting argument. — Wayfarer

Then show me were Wigner does more than I say - the arguments are there, but I think them of little merit. Choose one to discuss.

You simply referred to behaviourism, which has been obsolete since the 1940's. — Wayfarer

You know that is not an accurate rendering of Wittgernsteon's attitude towards mathematics. You can do better.

It's not clear to you. — Wayfarer

If it is clear to you what it is to be real in Platonic realism, you should be able to share. You will be aware that this is the most common criticism, so you presumably have a reply. Where can I find a real platonic circle?

What more could be said about the ontological status of abstracts other than that they are real abstracts? Analogously what more could be said about real physicals other than that they are real physicals? — Janus

It's a real physical object - it's not an illusion, it's not a reflection, its not a mirage...

It's a real abstract object - its not...?

I left out Frege. What is self-evident to one person may be rejected by another. In particular Frege's desire to develop a logical basis for arithmetic and hence for maths as a whole has been show impossible. Maths cannot be derived in its entirety from self-evident truths. — Banno

It is common knowledge that Frege's logicist project failed. Frege 0, Godel 1. And guess what?

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. 1 — Rebecca Goldstein

Besides, the point I was illustrating from Frege is that mathematical platonism is simply assumed by many logicians and philosophers, Frege being an example, and the illustrations of why he thinks that way are germane to the argument.

Where can I find a real platonic circle? — Banno

Platonic objects don't exist anywhere, they're purely noumenal, they're objects of thought. It's not that any physical circle is somehow imperfect, as many people say, but that the idea of a circle is precise and determinate, and would be so, regardless of whether any physical example exists or not.

So the question 'where can I find...' denotes a failure to grasp the point at issue. The whole point is, numbers and geometric forms and the like are transcendent, they are not located in space and time. And this is why naturalism has no way of conceiving or coping with them. Hence all those hysterical rhetorics I quoted from the Smithsonian article. If you admit the reality of transcendent objects then this poses grave difficulties for the standard-issue naturalism.

I know this is a very difficult point - I've been discussing this subject since day one on this and other forums. Conceiving of anything that doesn't exist in time and space requires a kind of 'through-the-looking-glass' realisation, a type of gestalt shift regarding what is real. Added to that, as i keep saying, and you keep ignoring, it goes against the grain of all of the accepted wisdom. We're conditioned to believe that what is real exists in time and space, and empiricism continually re-inforces that by insisting that anything considered real must be sense-able. Hence your question, 'where can I find it?'

Then show me were Wigner does more than I say... — Banno

Isaac Newton noticed that the path of a falling body (perhaps a thrown rock) on the Earth and the path of the moon in the sky are two particular cases of the more general notion of an ellipse. From this observation, he postulated the universal law of gravitation, which states that the gravitation between two objects is proportional to their masses. While Newton, given the restrictions of his day, could only verify the results with an accuracy of 4%, the law was later proved to be accurate to within less than a ten thousandth of one percent. This law, therefore, is a fantastic example of a mathematical formalism that has proved accurate beyond any reasonable expectations.

Quantum mechanics gives an even more astounding example. Matrix algebra had been studied independently of any applications by pure mathematicians for some time when Max Born realized that Werner Heisenberg’s rules of computation were formally identical with the rules of computation of matrices. Born, Pascual Jordan, and Heisenberg then replaced the position and momentum variables of Heisenberg’s equations of classical mechanics by these matrices and applied that result to an idealized problem. The new formulation worked, but would it work in a realistic setting, not just a toy problem? Within months, Wolfgang Pauli applied the new formulation to a realistic problem (a hydrogen atom) and the results matched up. Since Heisenberg’s original calculations were abstracted from problems that included the old theory of hydrogen atoms to begin with, this result was not too surprising. However, the “miracle” occurred next, when the matrix mechanics were applied to problems for which the Heisenberg rules no longer applied. — equations of motion of atoms with greater numbers of atoms. These observations were shown to agree with experimental data to within one part in ten million! Once again, mathematics developed independently of physics has been applied to physics to give spectacularly accurate results, far beyond the expectations of the original theory. 2.

So you may think that is mere 'sentimentalism' but I don't think it does it justice.

...the point I was illustrating from Frege is that mathematical platonism is simply assumed by many logicians and philosophers — Wayfarer

Yep - as I said,

...all this to say that Platonic Realism towards mathematical objects is not wrong, but just one way of treating those objects. It comes about by extending the notion of what is real to encompass numbers. — Banno

We might consider this alternate definition of real...

whatever is real is such as it is regardless of what anyone thinks about it — aletheist

On this account a painting is real, but it is also a construction. Our two views may not be mutually exclusive.

But i remain keen on the Austin's analysis of real as a term that gets its meaning by contrast, so again, It's a real abstract object - its not...?

you have to allow for the fact that not all 'intelligible objects' are real, because humans have imagination as well as intellect. Given imagination and intellect, all manner of synthetic abstractions can be constructed, but that doesn't mean that their are not real abstractions.

It is the ontological status of real abstractions that is at issue. Empiricism has to insist that they're the product of the mind, otherwise there's no conceptual space for them to be - they're not located in time and space, yet that is the entire theatre of operations for empiricism. This is the basis of the interminable argument about whether maths is real or invented. Of course, it's both, but at least in some fundamental respect, it's real. ("God created the integers....")

(I have to sign out for some hours, I supposed to be doing a self-training course, so have to drag myself away.)

So you may think that is mere 'sentimentalism' but I don't think it does it justice. — Wayfarer

Ok - what is the conclusion that you would like us to reach from this?

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?

(I have to sign out for some hours, I supposed to be doing a self-training course, so have to drag myself away.) — Wayfarer

Argh. You have my sympathy.

Empiricism has to insist that they're the product of the mind, otherwise there's no conceptual space for them to be — Wayfarer

Of course that is insufficient. It does not follow that they are real... and we still do not know what real is doing here.

Any argument is to be in suport of a sentiment felt by some mathematicians, that they are discovering rather than inventing. again, we need more than sentiment.

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? — Metaphysician Undercover

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.

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.

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.

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. — Metaphysician Undercover

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.

For instance in "I'm going for a walk", the only object denoted is "I". — Metaphysician Undercover

On the contrary, "going for" denotes a certain kind of relation as its object, and "a walk" denotes a certain kind of activity as its object. In fact, as symbols, words and phrases typically denote general concepts like these as their objects. The syntax of the sentence is what signifies the interpretant, which is the relation among the denoted objects that the corresponding proposition conveys.

And in your example of fictional writing, there are no objects denoted. The author simply builds up images of characters without denoting any objects. — Metaphysician Undercover

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.

It's a real abstract object - its not...? — Banno

It's a real physical object - it's not an illusion, it's not a reflection, its not a mirage...

It's a real abstract object - its not...? — Banno

...it's not an illusion, it's not imaginary, it's not a fiction. It's attributes are not a matter of opinion.

It's attributes are not a matter of opinion. — Janus

Ah. Is this the possibility that bothers you?

If maths were invented, do you think it might be subject to mere opinion?