What do you mean by "invented"? Invented as opposed to what?

Discovered.

I don't think there is a clear enough distinction between invented and discovered in this context. Objects, places, substances and so on in the world may unequivocally be said to be disovered

Games like chess or backgammon may unequivocally be said to be invented.
Games are arbitrary, though, and lack real world applications beyond themselves.

On the other hand, was the internal combustion engine invented or discovered? It could be said that it represents a possibility that was discovered, someone else might say it was invented.

Mathematics has real world applications. I think it's fair to say that quantity and the possibility of counting is discovered, not invented; even some animals can recognize number to varying degrees.

Proofs in mathematics are said to be discovered, as they are logical possibilities that arguably would exist even if no one discovered them.

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

Didn't Russell rather skewer that approach? (On Denoting.)

And in your example of fictional writing, there are no objects denoted. — Metaphysician Undercover

Speaking literally, yes.

The author simply builds up images of characters — Metaphysician Undercover

This needs clarifying. Produces strings of alphabetic characters? Sure. Conjures mental images of non-existent and hence undenoted people? Well, "mental" is problematic but let's park that. Or skirt it, by assuming the book is a picture book. Goodman's very neat solution is then to read "images of characters" e.g. "picture of Pickwick" not as requiring two separate denotata, a picture and a Pickwick, but as long (if only slightly) for "Pickwick-picture", a one-place predicate applying to a certain sub-class of pictures.

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. — aletheist

If what version were true? If my suggested version, then why (could we) not?

The possible objection that we must first understand what a man or a unicorn is to know how to apply "man-picture" or "unicorn-picture" seems to me quite perverted. We can learn to apply "corncob pipe" or "staghorn" without first understanding, or knowing how to apply, "corn" or "cob" or "corncob" or "pipe" or "stag" or "horn" as separate terms. And we can learn, on the basis of samples, to apply "unicorn-picture" not only without ever having seen any unicorns but without ever having seen or heard the word "unicorn" before. Indeed, largely by learning what are unicorn-pictures and unicorn-descriptions do we come to understand the word "unicorn"; and our ability to recognize a staghorn may help us to recognize a stag when we see one. — Goodman, Languages of Art

Proofs in mathematics are said to be discovered, as they are logical possibilities that arguably would exist even if no one discovered them. — Janus

I think it is natural to divide subjects between what is 'in the mind' and what is 'in the world'. What is 'in the world' is said to exist independently of any act of observation on the human's part - the vast universe with all of billons of galaxies, this earth and its solar system and so on.

In contrast, what is 'in the mind' is thought to comprise language, culture, reason, concepts, mathematics, science, the whole furniture of thought.

So within this mindset it is natural to presume that mathematical proofs are 'in the mind'. Where else, it is asked, could they be? They're not 'out there somewhere'. But if they're in the mind, then how could they be discovered, because they're not objectively existent? If they're in the mind, they must be a product of the mind. They are, it is said, 'mind-dependent' and so categorically different to the supposedly 'mind-independent' domain of the vast natural world.

This is where I think the conflict between mathematical platonism and empiricism lies. Many of the sources I quoted in this thread show that this is exactly how the conflict is seen by empiricists. I am not 'attacking staw men' here.

(My view is that realiity is experienced reality and that this even goes for the hard sciences. Nothing exists truly independently of any act of observation, or rather, whatever we think or say about 'what exists' implicitly implies the act of observation, which provides the conceptual framework within which any and all existence statements are made. Not seeing this is precisely the meaning of 'the blind spot' which had a very hostile reception when I mentioned it in June 2019.)

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?

Didn't Russell rather skewer that approach? (On Denoting.) — bongo fury

Thus "the present King of France," "the round square," etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This in itself is a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe on the law of contradiction. It is contended, for example, that the existent present King of France exists, and also does not exist; that the round square is round, and also not round; etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred. — Russell - On Denoting

Here Russell gets Meinong wrong. Meinong does not claim that such objects exist. He claims that some objects exist, some subsist and some don't exist at all.

Well that makes everything perfectly clear.

Proofs in mathematics are said to be discovered, as they are logical possibilities that arguably would exist even if no one discovered them. — Janus

A popular science-math weekly used to offer challenges for readers to submit original proofs for mathematical theorems. The Pythagorean theorem received about a hundred different proofs from creative readers.

Ha. Well at the very least is shows that Russell didn't skewer Meinong's position. He didn't even address it.

I think it is natural to divide subjects between what is 'in the mind' and what is 'in the world'. What is 'in the world' is said to exist independently of any act of observation on the human's part - the vast universe with all of billons of galaxies, this earth and its solar system and so on. — Wayfarer

That kind of dualistic thinking does seem to be "natural". I would rather more simply say there is a distinction between what exists and is what it is independently of opinion and what doesn't exist independently of opinion and is only what it is on account of opinion.

(My view is that realiity is experienced reality and that this even goes for the hard sciences. Nothing exists truly independently of any act of observation, or rather, whatever we think or say about 'what exists' implicitly implies the act of observation, which provides the conceptual framework within which any and all existence statements are made. Not seeing this is precisely the meaning of 'the blind spot' which had a very hostile reception when I mentioned it in June 2019. — Wayfarer

I agree that our reality is experienced reality, but we can easily enough imagine that there are realities that exist independently of our experiencing them. I know, you will say 'It is still us imagining them'; but I don't see that as a cogent objection since I don't think the purported contradiction said to be inherent in that holds.

Of course we can't experience realities which are beyond our experience, that would be a contradiction, but we can imagine them as existing, and we can (rightly, in my view) think it is likely that they do exist. On the contrary it seems absurd to say, for example, that nothing existed prior to the advent of humans.

Nice example!

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? — Metaphysician Undercover

This question just confirms an ongoing failure (or refusal) to understand the technical definitions of denotation and signification within semeiotic.

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. — Metaphysician Undercover

That is not what I said. There is no such thing as pure signification (without denotation) in common usage. Likewise, there is no such thing as pure denotation (without signification) in common usage. Instead, in practice every sign both denotes its object (what it stands for) and signifies its interpretant (what it conveys about that object). This is most readily evident in a proposition, where the subjects (terms as names) denote the objects and the predicate (embodied as syntax) signifies the interpretant. The fundamental principle of semeiotic (following Peirce) as distinguished from semiology (following Saussure) is that a sign thus stands in an irreducibly triadic relation with its object and its interpretant, rather than there being only a dyadic relation between signifier and signified.

Proofs in mathematics are said to be discovered, as they are logical possibilities that arguably would exist even if no one discovered them — Janus

It's the theorem that's discovered/created first. Then the search for a proof. Math is not just challenging others to solve a stated problem, although for many that is a competitive aspect highly desirable.

Good correction: that makes sense to me.

That kind of dualistic thinking does seem to be "natural". — Janus

The duality of ‘self-and-world’ develops in infancy. Interestingly, in translations of the early Buddhist texts, the expression ‘self and world’ is sometimes used in places where you would expect ‘self’ - the connotation being that self and world are mutually interdependent or in modern terms 'co-arising'. I think that since the Enlightenment, philosophy has increasingly lost sight of this interdependence. (Arguably, it was the intuition of interdependence which was the basis of Berkeley's philosophy.)

I would rather more simply say there is a distinction between what exists and is what it is independently of opinion. — Janus

It was Plato who disparaged ‘mere opinion’ masquerading as knowledge. Scientific method is ostensibly aimed at attaining mathematical certainty with respect to its objects of analysis. The problem is, as we have discussed many times, it does so at the cost of eliminating the qualitative dimension of existence without which human life is meaningless. This is a consequence of the division of the objects of knowledge according to the primary and secondary attributes, originating with Galileo, which gave rise to what Whitehead described as the 'bifurcation of nature'.

it seems absurd to say, for example, that nothing existed prior to the advent of humans. — Janus

I question that any notion of existence is coherent absent a subject to whom it is meaningful. This goes for the most rigorous scientific modelling of the Cosmos also. The units of measurement are meaningful to humans, and the human mind furnishes the perspective within which statemements about time, space, duration and distance are meaningful.

The problem of including the observer in our description of physical reality arises most insistently when it comes to the subject of quantum cosmology - the application of quantum mechanics to the universe as a whole - because, by definition, 'the universe' must include any observers. Andrei Linde has given a deep reason for why observers enter into quantum cosmology in a fundamental way. It has to do with the nature of time. The passage of time is not absolute; it always involves a change of one physical system relative to another, for example, how many times the hands of the clock go around relative to the rotation of the Earth. When it comes to the Universe as a whole, time loses its meaning, for there is nothing else relative to which the universe may be said to change. This 'vanishing' of time for the entire universe becomes very explicit in quantum cosmology, where the time variable simply drops out of the quantum description. It may readily be restored by considering the Universe to be separated into two subsystems: an observer with a clock, and the rest of the Universe. So the observer plays an absolutely crucial role in this respect. Linde expresses it graphically: 'thus we see that without introducing an observer, we have a dead universe, which does not evolve in time', and, 'we are together, the Universe and us. The moment you say the Universe exists without any observers, I cannot make any sense out of that. I cannot imagine a consistent theory of everything that ignores consciousness...in the absence of observers, our universe is dead'.

(Paul Davies, The Goldilocks Enigma: Why is the Universe Just Right for Life, p 271)

The duality of ‘self-and-world’ develops in infancy. — Wayfarer

I wouldn't cal it "self and world" but rather 'self and other'. Even animals arguably have this sense. But it is this very sense and acknowledgement of existences other than our own that underpins the coherence of imagining existences that are independent of our experiencing of them, our possibility of experiencing them or even of our existence tout court.

So, I cannot agree with this:

I question that any notion of existence is coherent absent a subject to whom it is meaningful. — Wayfarer

This is an ambiguous statement. Of course a notion of existence can only be coherent to a subject to whom it is meaningful; that is tautologically true on one reading. But you seem to be wanting to extend that to say that we cannot coherently imagine that there are existences that are completely independent of us; and with that I disagree.

It may be incoherent for you, on account of its not cohering with your particular presuppositions, but it presents no problem for me, because my presuppositions are consistent with it.

In fact I would say that it is incoherent to imagine that nothing existed prior to the advent of humans. I would say that it is impossible to coherently imagine that because it flies in the face of all our knowledge.

Also, I'm not convinced of the relevance of the so-called "observer problem" in QM, because there is no uncontroversial interpretation of the significance of that problem for ontology. And in any case by your own lights QM is just another human-based investigation and so can tell us nothing about anything beyond human experience.

It's the theorem that's discovered/created first. Then the search for a proof. Math is not just challenging others to solve a stated problem, although for many that is a competitive aspect highly desirable. — jgill

Seems to me that proofs can neither be independently created nor discovered. Even together, creativity with serendipitous discovery aren't sufficient to make an Euler. Something is still missing. Then there is the issue of computer generated proofs. What kind of thinking is involved there?

That is not what I said. There is no such thing as pure signification (without denotation) in common usage. Likewise, there is no such thing as pure denotation (without signification) in common usage. Instead, in practice every sign both denotes its object (what it stands for) and signifies its interpretant (what it conveys about that object). This is most readily evident in a proposition, where the subjects (terms as names) denote the objects and the predicate (embodied as syntax) signifies the interpretant. The fundamental principle of semeiotic (following Peirce) as distinguished from semiology (following Saussure) is that a sign thus stands in an irreducibly triadic relation with its object and its interpretant, rather than there being only a dyadic relation between signifier and signified. — aletheist

Sure there is pure signification, in the case of any abstract use, a universal, like "temperature", "big", "good", "beauty" "green", "wet", and the list goes on and on. You just want to insist that these can only be used when describing an object, to support your special form of Platonism. But it's not true. We use all these terms as a subject when we say things like "temperature is a measurement", "big is a size", "good is desirable", beauty is what the artist seek", "green is a colour". These are phrases of pure signification, and to turn the subject into an object will most probably lead to category mistake. Because then we lose the capacity to distinguish between a physical object denoted, and a subject of study denoted. If these two are the same, as "object denoted", category mistake will prevail.

Sure there is pure signification, in the case of any abstract use, a universal, like "temperature", "big", "good", "beauty" "green", "wet", and the list goes on and on. — Metaphysician Undercover

No, these words are examples of signs whose objects--that which they denote--are general concepts.

We use all these terms as a subject when we say things like "temperature is a measurement", "big is a size", "good is desirable", beauty is what the artist seek", "green is a colour". These are phrases of pure signification — Metaphysician Undercover

No, in each of these propositions there are two or three subjects denoting two or three objects, and the interpretant conveys something about the logical relation between those objects. The interpretant of each individual word is the aggregate of all the different propositions that include it, which we attempt to summarize whenever we write a definition of it.

Because then we lose the capacity to distinguish between a physical object denoted, and a subject of study denoted. If these two are the same, as "object denoted", category mistake will prevail. — Metaphysician Undercover

The category mistake is conflating different definitions of "object" and "subject" that apply in different contexts. An object is not necessarily something physical, and a subject is not necessarily something that we study. In semeiotic, an object is whatever a sign denotes, and a subject is a term within a proposition that denotes one of its objects.

Something is still missing — magritte

Intuition, instinct, imagination, et al.

Then there is the issue of computer generated proofs. What kind of thinking is involved there? — magritte

I suspect ingenuity on the part of the program creator. After that a computer does what a computer does. Here's a good article: Computer Proofs

From which:

A proof is strange, though. It’s abstract and untethered to material experience. “They’re this crazy contact between an imaginary, nonphysical world and biologically evolved creatures,” said the cognitive scientist Simon DeDeo of Carnegie Mellon University, who studies mathematical certainty by analyzing the structure of proofs. “We did not evolve to do this.”

:up:

No, these words are examples of signs whose objects--that which they denote--are general concepts. — aletheist

I know that's what you think, but I disagree. I think that you're way of looking at things creates ambiguity in the meaning of "object", which leads to equivocation between mental objects and physical objects, resulting in category mistake.

The interpretant of each individual word is the aggregate of all the different propositions that include it, which we attempt to summarize whenever we write a definition of it. — aletheist

Oh, come on. One cannot expect to consider all the different propositions that include a word, when interpreting that word. Many could be inconsistent or contradictory. In a logical proceeding there are stipulated propositions. But there are no objects, just subjects. The propositions make predications of subjects, not objects.

An object is defined according to the law of identity, as unique, primary substance, but a logical subject is not unique, as secondary substance. So the relations you refer to may be applied to multiple objects, as universals, because a subject is not limited to representing one object. But in calling these subjects "objects" you imply the uniqueness of an object, as required by the law of identity, Therefore in defining "object" in this way you loose the capacity to distinguish between whether the relation referred to is a unique relation, specific to a particular situation (object in my sense), or a universal relation, common to numerous objects (in my sense), represented by a single subject. In other words, by calling the subject an object, we loose the capacity to distinguish uniqueness, due to the ambiguity and the category mistake which will prevail.

An object is not necessarily something physical, and a subject is not necessarily something that we study. In semeiotic, an object is whatever a sign denotes, and a subject is a term within a proposition that denotes one of its objects. — aletheist

Do you believe in the law of identity and the uniqueness of an object? If so, then how can you allow that "whatever a sign denotes" is an object, when the same sign denotes different things in the minds of different people? Do you assume an independently existing Idea, or Form, as the ideal conception, or unique object denoted, separate from the less than perfect ideas of individual human beings, which are all slightly different?

I know that's what you think, but I disagree. — Metaphysician Undercover

I know that you disagree, and at this point I find that comforting.

An object is defined according to the law of identity, as unique, primary substance, but a logical subject is not unique, as secondary substance. — Metaphysician Undercover

There you go again, making stuff up to sound knowledgeable. That is not how "object" and "subject" are defined within semeiotic. I get it, you reject those definitions, so again we can stop wasting each other's time.

The question though, do you see that I have very good reason to reject those definitions? They increase ambiguity, leading to equivocation and category mistake. In the interest of understanding, you ought to reject them as well. Don't you think?

On the contrary, within semeiotic the definitions of terms including "object" and "subject" are unambiguous and foster greater understanding. As far as I can tell, your only reason for rejecting them is that they are different from your preferred definitions, which you want to impose on any and every context.

These mathematical axioms require that a term signifies an object. Only Platonism can support this prerequisite. — Metaphysician Undercover

One example, please. Or if it's above, kindly point me there.

I am a little moody and out of sorts, so I apologize that I made no attempt to reply. I will give my brief account of Benecerraf's argument, from what I could surmise so far.

It is key that the author specifically distinguishes knowledge from theory. The first conjecture he makes is that mathematical knowledge is obtained by experiencing such situations that produce belief (causally). Assuming that I interpret the paper correctly, the belief is expressed extensionally, as the potential evidence that would be considered in agreement or disagreement with the experiencing subject. Whether the belief is justified or not in some communicated sense is not relevant, because it is defined by the circumstances that compel it. Although, this point is developed at a much higher level then what is necessary for me to expound its empirical meaning, particularly what solicits some persuasion and how it is embodied, it appears to be deliberately extensionally situated and not intensionally represented. Anyway, the argument is made, I think, that truth in abstract theories is evaluated in terms of the congruency between the conclusions and the discovered facts. In other words, we synthesize theories intensionally and then check their validity by observing situations that agree with their prerequisites and we make determination if the inferred theorems match with the states of affairs, according to some rational interpretation (i.e. Tarski's interpretation). (If the theory is logically consistent and we always observe situations that meet its prerequisites part way, we could speculate that rational logic is not good qualitative control.) If I have understood the conclusion correctly, since we have no limitations imposed on how we synthesize abstractions intensionally, and we have no analytic correspondence between our state of belief and the extension of the facts which will compel us to believe or dissuade us from believing, abstractions cannot be explained as beliefs. In other words, theoretic intensions and persuasion extensions cannot be matched apriori.

(Made an edit to express the conclusion in what I consider slightly better terms.)

Analogy,
My umbrella is designed to guard me against rain. Does it therefore rain if I have opened my umbrella? It needn't be. I might be checking to see if it works. (The object is designed as such, but my action of raising it is ultimately independent of my need for it). I run towards the visor of a building with newspaper over my head. Is it raining? The object is not designed as such, but under the present circumstances, my behavior with it indicates that I am in need of cover (my behavior is a belief indicator).

It is a rather difficult text and I am not on that same level. The author makes reference to another paper, "Platonism and the Causal Theory of Knowledge" by Mark Steiner. If someone has access to it and could elaborate its content, it might shed light.

Thanks. My main source for Benecareff's paper is the article The Indispensability Argument in the Philosophy of Mathematics. It makes many straightforward claims about why mathematics poses a problem for empiricism:

Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to debar any knowledge of mathematical objects. ....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. ....Some philosophers, called rationalists, claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

So, there's really an obvious conflict here, between what has always been understood as 'reason', and what today's 'best epistemic theories' are. I simply conclude from all this that today's 'best epistemic theories' are deficient in some fundamental respect, in not being able to accomodate the faculty which is uniquely human, namely, reason. And that's because we're not simply physical creatures, whose capacities for learning are exhausted by our physical bodies. So if it's one or the other, I choose reason.

One example, please. — tim wood

Set theory, and the axiom of extensionality.

As far as I can tell, your only reason for rejecting them is that they are different from your preferred definitions, — aletheist

I conclude that you didn't read, or for some reason couldn't understand what I said then.

and set theory, to begin with. These mathematical axioms require that a term signifies an object. Only Platonism can support this prerequisite. — Metaphysician Undercover

One example, please.
— tim wood

Set theory, and the axiom of extensionality. — Metaphysician Undercover

Ok. You make a claim: "These mathematical axioms require that a term signifies an object. Only Platonism can support this prerequisite." Eh? What does that even mean? That the axioms "require" something to be axioms? Or that as axioms they mandate something? I'm not finding sense here.

What axioms, what objects? Just a simple example ought to suffice to demonstrate the necessity of Platonism.