This point is interesting. — javi2541997

The essay itself is interesting. You can find a copy here. Frege was not at all like Berkeley, from what I can see - more cautious, not prone to sweeping statements or grand metaphysics.

I explained why a rational empiricist can be possibilian which could allow this discussion to progress further, without being divided between people of different persuasions. I made the case, epistemically justified, that possibilianism is natural consequence of rational empiricism, for philosophical subjects which lack clear determination. — simeonz

I'm not familiar with this term 'possibilian' and I can't find it on the Web. Perhaps you could explain?

But, beyond that, I fear we're talking past one another. Let's go back to the beginning of your first post in this thread:

The question can't be answered without some kind of speculation, obviously. — simeonz

Which 'question' do you mean? What question do you think I'm posing that 'can't be answered without some kind of speculation?'

there are certain areas in physics where math, as they claim, "breaks down" e.g. black holes, the Big Bang singularity, to name a few. — TheMadFool

Yes, true, although black holes, in particular, were theoretically posited as a direct consequence of Einstein's laws, weren't they? And again I'm not claiming that science is all-knowing even in principle. Look carefully at the OP again. The topic I'm interested in is: are numbers real? And if so, in what sense are they real?

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.

'm not familiar with this term 'possibilian' and I can't find it on the Web. Perhaps you could explain? — Wayfarer

Possibilianism is mostly concerned with theistic claims, but it is essentially attitude open towards the exploration of unproven claims, as long as they are suggested hypothetically.

Which 'question' do you mean? What question do you think I'm posing that 'can't be answered without some kind of speculation?' — Wayfarer

The question about the nature of experience. What explains having experience and hence knowledge. How does it form - does experience emerge from the innate ability of the material substance to be self-cognizant of its configuration (panpsychism, emergent materialism, pantheism), does it emerge by virtue of connection to higher cognitive self (substance dualism), does it emerge as creative fictional introspection (idealistic existential monism), or does it emerge through collaborative enactment (enactvism). Those are some ideas. All of those ideas explain science in different manner and with somewhat different consequences. Some justify the attitude of empiricism completely, and others explain this attitude, but do not justify it.

Do you agree that abstractions "in the mind" can be formed differently according to each of those hypotheses?

The essay itself is interesting. You can find a copy here. — Wayfarer

Thank you for sharing it with me. I am going to check it out.

each of which hypotheses?

I will try to briefly summarize my take on each idea. Those are my interpretations, so you could argue with them or not, but I am not sure that any of them coincide completely with Spinoza, or Leibniz, or some established authority.

Panpsychism proposes that nature is self-cognizant. The problem is how do the constituents of matter enact mental coherence and unity between each other, to form singular consciousness. But if some kind of emergentistic view applies, then matter simply perceives matter and captures knowledge and ideas by representation. This representation automatically traps the abstractions that homomorphically approximate the environment (also panpsychic matter). Pantheism is very close, but emergentism follows more naturally, because the deity is capable of self-awareness (possibly unrelatable to our understanding) and the material constituents just capture local fragmented perspectives of it.

Substance dualism keeps the ideas in the mind separate from the physical world. The brain and the world are a computational device. The mind is the actual carrier of experience. The problems that it faces are - where and how is the mind created, is the knowledge captured in a supervening manner preserved after the physical embodiment is disassociated, what happens when the body is mentally impaired or the amount of bodies decrease, etc.

I may be coining the term idealistic existential monism, but the idea is that our identity is fictional and our experience of the world happens within the confines of single cognizant entity that also creates it. (Edit: We are the split personalities of a deity that willingly experiences disassociative identity disorder. This is similar to pantheism, but doesn't require allocation of cognizant potential to matter. In fact, I see no apparent obstruction to creating and destroying identities in this hypothesis, because they serve only as epistemic device to the creator.) The problem here is primary motivation, but cognition and understanding is trivially possible.

Enactivism is as close to what was referred on the forum as intersubjective idealism. Each person is a creative force, but the collective effort is contingent and self-regulating. How it self-regulates constitutes the challenge for this hypothesis.

Edit. Obviously, you are free to amend and adapt my take on the ideas or propose your own.

I see. Actually I read your posts slightly out of order, I had not noticed the reply you gave above this one, when I asked the question ‘each of which hypotheses’?

You’re clearly a deep and well-read thinker about these questions, but I still feel that in respect of this particular thread, we’re talking at cross-purposes. The rhetorical question I’m posing is, why is mathematical Platonism out of fashion?Why is it that many serious mathematicians and philosophers seek to discredit it, and to explain our ability to mathematise in naturalistic or reductionist terms? It’s really rather a specialised question, and one I am barely qualified to consider, considering how technical many of the arguments are. But it’s a question I’m intrigued by, in fact my first forum post in 2010 was about this very question, and to me it’s still very much a live issue.

Enactivism is as close to what was referred on the forum as intersubjective idealism. Each person is a creative force, but the collective effort is contingent and self-regulating. How it self-regulates constitutes the challenge for this hypothesis. — simeonz

I’m very much attracted to that idea. I find the key figure there is Husserl. His ideas of the umwelt and lebenswelt, as kind of ‘meaning-environments’, speaks volumes to me.

The rhetorical question I’m posing is, why is mathematical Platonism out of fashion?Why is it that many serious mathematicians and philosophers seek to discredit it, and to explain our ability to mathematise in naturalistic or reductionist terms? It’s really rather a specialised question, and one I am barely qualified to consider, considering how technical many of the arguments are. — Wayfarer

Yes. In retrospect, I realize that I got a little carried away from the topic. I got stuck on the issue of the nature of experience. I'll think about whether I can research and contribute something more topical later (edit:...or much later).

no probs I always appreciate your posts and the frankness of your approach. :up:

Yes, true, although black holes, in particular, were theoretically posited as a direct consequence of Einstein's laws, weren't they? And again I'm not claiming that science is all-knowing even in principle. Look carefully at the OP again. The topic I'm interested in is: are numbers real? And if so, in what sense are they real?

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

Well, what exactly does "real" in "numbers are real" mean? I'm no mathematician but I can say with some degree of confidence that numbers are, at the end of they day, abstractions - they are, in the most basic sense, patterns in sets: The number 1 is the pattern in the sets {0}, {a}, {red}, {fox} and the number 3 is the pattern in the sets {good, 3, pee}, {%, fee, bravery}, {love, dog, +} and so on. The question then is, are abstractions real? Finding an answer to this question is the first order of business, no? So, what do you think? Are abstractions real?

A good starting point is to realize that all it takes is a mind sensitive to patterns, with a pattern-recognition module so to speak, to see that there is a pattern in the world as we know it that humans have named numbers. Any system capable of detecting patterns will, sooner or later, hit upon the idea of number from its observations of the world. Doesn't that indicate that though it takes a mind or a pattern-detecting system to conceive of numbers, the pattern has to exist outside of the mind, in the world "out there" as opposed to "inside our minds"? How can the mind perceive of something that doesn't itself exist in some sense of that word? Beats me.

Have you also looked into Imaginary Numbers? The square root of -1, according to mathematicians, doesn't exist and that means, the aptly named, real numbers exist. How exactly are mathematicians using the words "exist" when making statements about the reals compared to imaginary numbers?

The square root of -1, according to mathematicians, doesn't exist and that means, the aptly named, real numbers exist. — TheMadFool

I don't think this is right. Imaginary numbers exist as mathematical entities used by mathematicians. In exactly the same way that the real numbers exist for mathematicians: they exist because they are used. The imaginary numbers are just plotted on a different number line (not the real axis).
The term imaginary number is considered to be a misnomer by many mathematians.

I don't think this is right. Imaginary numbers exist as mathematical entities used by mathematicians. In exactly the same way, the real numbers exist for mathematicians: they exist because they are used.
The term imaginary number is considered to be a misnomer by many mathematians. — emancipate

@Wayfarer Corrigendum.

By the way, if imaginary numbers exist, what is the square root of -1? I know the square root of 4 is 2, a number; I know the square root of 2 is 1.414..., another number.

All real numbers are (probably?) instantiated in the universe. Take for example pi, wherever you see something circular/spherical, it's there as the ratio between circumference and diameter.

Where is a real-world instantiation of the square root of -1? Electronics? Semiconductors?

By the way, if imaginary numbers exist, what is the square root of -1? I know the square root of 4 is 2, a number; I know the square root of 2 is 1.414..., another number. — TheMadFool

They exist as abstractions, mathematical concepts.

All real numbers are (probably?) instantiated in the universe. Take for example pi, wherever you see something circular/spherical, it's there as the ratio between circumference and diameter. — TheMadFool

There are certainly mathematical abstractions without a concrete actualisation, and these include real numbers. I think uncomputable real numbers would be an example (chaitins constant). Or a number a billion times larger than the atoms of the observable universe. I mean, there is always a larger number in abstract land..

They exist as abstractions, mathematical concepts. — emancipate

Yet real numbers and "imaginary" numbers aren't exactly like each other. I can easily express any real number with the numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and a decimal point but not an "imaginary" number. This difference, how does it impact the OP's concerns?

You seem to have missed the point but it's entirely my fault. What I meant was real numbers - as a category - has real-world instantiations i.e. there's at least one real number that's actually, let's just say, measurable with a straight edge. Not so for "imaginary" numbers.

I don't want to derail the thread and must be honest I am not a mathematician. However, it's my understanding that in your example numbers are not the thing being measured by the straight edge. The numbers are themselves a measure, or quantity of an arbitrary unit. Complex numbers can be used to measure quantites of other kinds (not straight lines). That means they have real world applications. I don't know the specifics but someone with real mathematical knowledge can back me up :D

Edit: The strength of an electromagnetic field can be measured by complex numbers (apparently).

Math ain't my cup of tea and I'm dangerously close to, as Neil deGrasse Tyson puts it, "...the perimeter of my ignorance...". I suppose I should know when to quit. G' day!

I'm not exactly in my element either. Blind leading the blind here.

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.

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. — Metaphysician Undercover

Yep. Sir Paul Davies is at least one of many theoretical physicists who would agree with that! :up:

I think that since the success of the nominalist attitude, which was one of the main forerunners of empiricism generally, that scholastic realism has been forgotten to such an extent that there is barely any awareness of what it meant. — Wayfarer

:up:

Where is a real-world instantiation of the square root of -1? — TheMadFool

$i$ appears in the Schrodinger equation:

$i\hbar\frac{\partial}{\partial{t}}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle$

Well, what exactly does "real" in "numbers are real" mean? — TheMadFool

That is a historical question, and my understanding is that mathematicians refer to "real" numbers only to distinguish them from so-called "imaginary" numbers; the latter term actually came first.

The question then is, are abstractions real? — TheMadFool

In general philosophy, "real" means being such as it is regardless of what anyone thinks about it. In philosophy of mathematics, realism ascribes this nature to mathematical objects, including both real and imaginary numbers.

Where is a real-world instantiation of the square root of -1? Electronics? — TheMadFool

Yes, it is routinely used in circuit analysis and design.

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

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. Also, as I have explained elsewhere, signs denote their objects; what they signify are their interpretants.

Banno - as discussed. — Wayfarer

Cheers. Not quite the direction I thought you had in mind, but that will make it more interesting. I will do some reading.

A partial digest:

https://plato.stanford.edu/entries/platonism-mathematics/
https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
https://www.smithsonianmag.com/science-nature/what-math-180975882/

Well, what exactly does "real" in "numbers are real" mean? — TheMadFool

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. I think this is why platonic realism is so firmly resisted - it cuts against the grain of naturalism.

I don’t think this includes imaginary numbers or fictitous entities (like Sherlock Holmes or Bugs Bunny). Once mathematics becomes thinkable, then all kinds of fictionalised and imaginary systems can be invented, but I don’t think it detracts from the point at issue.

I think it what platonic realism does include is an astonishingly wide range, however. Think of the 'elements of reality' that are likewise only intellectual in nature, but are nevertheless real: 'interest rates, mortgages, contracts, vows, national constitutions, penal codes and so on. Where do interest rates "exist"? Not in banks, or financial institutions. Are they real when we cannot touch them or see them? We all spend so much time worrying about them - are we worrying about nothing? In fact, I'm sure we all worry much more about interest rates than about the existence or non-existence of the Higgs boson! Similarly, a contract is not just the piece of paper, but the meaning the paper embodies; likewise a national constitution or a penal code’ ~ Neil Ormerod.

The human 'meaning-world' is constituted from these elements, but they’re not 'out there somewhere'. They don't exist in the same way that flowers or pens or chairs exist but are real nonetheless.

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 human 'meaning-world' is constituted from these elements, but they’re not 'out there somewhere'. They don't exist in the same way that flowers or pens or chairs exist but are real nonetheless. — Wayfarer

It seems to me that number is obviously real. There is a real concrete difference between two apples and three apples. Numerals also are obviously real, they can be written and spoken. I can't see what all the controversy is about if one side (the platonic realists) are not arguing that numbers as actual entities of some kind "exist somewhere" (although obviously not in spacetime) or at least somehow.

The question for them is where or how do numbers as actual independent entities exist, or what would it mean to say they are real beyond saying that they are (real) abstractions?

To justify calling an imaginary thing "an object" requires some form of Platonism. — Metaphysician Undercover

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. No form of Platonism is required to affirm this.

It seems to me that number is obviously real. — Janus

Not if you subscribe to fictionalism.

The question for them is where or how do numbers as actual independent entities exist, or what would it mean to say they are real beyond saying that they are (real) abstractions? — Janus

The whole debate is exactly about the ontological status of abstracts. Mathematical Platonism says that the real numbers are not dependent on any mind, i.e. they’re not the product of thought. As Frege puts it, that they are grasped by the mind ‘in the same way the hand grasps a pencil’. But if they’re real, then what kind of existence do they have? What does it mean to say abstract objects exist? Where are they? This then circles back to - why, they exist ‘in the mind’.

If you glance at those refs I provided to Banno above, these arguments are discussed. Last one is the best starting point.