In short, it isn't obvious that mathematical platonism necessitates a commitment to only one construal (one use of ∃) of what it means to exist. — J

From here, "[p]latonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices."

I'm not sure how this can be further distinguished. Either some set is a mind-independent abstract object or it's not.

The question was much more ordinary: What are the concrete contents or data of which Husserl speaks, that allow us to form our idealization of numbers? Can you give an example of how this might work? — J

Here’s my long-winded attempt at a Husserlian explanation of the subjective constitution of number:

In Philosophy of Arithmetic(1891), Husserl described a method for understanding the constitution of a multiplicity or plurality composed of independent parts, which he dubbed ‘collective combination'. According to Husserl, the basis of any sort of whole of independently apprehended parts(a whole in the pregnant sense) is the collective combination, which is an abstracting act of consciousness uniting parts.

“Collective combination plays a highly significant role in our mental life as a whole. Every complex phenomenon which presupposes parts that are separately and specifically noticed, every higher mental and emotional activity, requires, in order to be able to arise at all, collective combinations of partial phenomena. There could never even be a representation of one of the more simple relations (e.g., identity, similarity, etc.) if a unitary interest and, simultaneously with it, an act of noticing did not pick out the terms of the relation and hold them together as unified. This 'psychical' relation is, thus, an indispensable psychological precondition of every relation and combination whatsoever.”

In any such whole the parts are united in a specific manner. Fundamental to the genesis of almost all totalities is that its parts initially appear as a temporal succession.

“Succession in time constitutes an insuppressible psychological precondition for the formation of by far the most number concepts and concrete multiplicities - and practically all of the more complicated concepts in general.”“Almost all representations of multiplicities - and, in any case, all representations of numbers - are results of processes, are wholes originated gradually out of their elements. Insofar as this is so, each element bears in itself a different temporal determination.””Temporal succession forms the only common element in all cases of multiplicity, which therefore must constitute the foundation for the abstraction of that concept.”

The first step of constitution of a multiplicity is the awareness of the temporal succession of parts, each of which we are made aware of as elements “separately and specifically noticed”. In the case of numbers, one must abstract away everything else about those elements (color, size, texture) other than that they have been individually noticed as an empty ‘unit’. The concept of number is only possible once we invent the idea of identical sameness over time ( same thing, different time). This concept is not derived from the concrete data of experience (i.e. real apples as their appearance is given to us via continually changing perspectives). Rather, it is a concept we impose upon a world in continual flux. It was necessary to invent the concept of identity, and its pure repetition, in order to have the notion of the numeric unit.

The collective combination itself only emerges from a secondary act of consciousness. This higher order constituting sense changes what was originally a temporal succession into a simultaneity by ‘bringing' back ‘ the previous parts via reflecting on them in memory. Husserl says that a combination of objects is similar to the continuity of a tone. In both cases, a temporal succession is perceived through reflection as a simultaneity.

“For the apprehension of each one of the colligated contents there is required a distinct psychical act. Grasping them together then requires a new act, which obviously includes those distinct acts, and thus forms a psychical act of second order.” “It is essential that the partial representations united in the representation of the multiplicity or number be present in our consciousness simultaneously [in an act of reflection].”

The constitution of an abstract multiplicity is analogous to the creation of any whole, even though the former involves a peculiarly external form of unification in comparison to combinations unified by similarity or continuity.

A key feature of the fact that a totality is a product of a temporally unfolding series of sense acts is that prior elements of the originally apprehended series have already changed by the time we move on to the succeeding elements of that series. “In forming the representation of the totality we do not attend to the fact that changes in the contents occur as the colligation progresses.” The secondary sense-forming act of the uniting of the pasts into the whole is not, then, ‘faithful' to the original meaning of the parts it colligates, in that they have already changed their original sense via the passage of time at the point where we perform the uniting act of multiplicity. Rather than a being faithful, the sense of the unification act may better be described as a moving beyond the original sense-constituting acts forming the apprehension of the parts. In forming a new dimension of sense from retentional and protentional consciousness, the unifying act of totalization idealizes the parts that it unifies. In addition to the abstractive concept of groupness (collective combination), many kinds of more intimate idealizations are constituted as wholes out of original temporal successions. We can see this clearly in the case of the real object, an ideal totality formed out of a continuous synthetic flow of adumbrations in which what is actually experienced in the present is not the ‘faithful', that is, actual presencing of temporally simultaneous elements but a simultaneity of retentional series, present sense and protentional anticipations.

In a way, the number 5 implies all other numbers, because its meaning is rooted in its place in a sequence. And every thing is like that.

↪Joshs
In a way, the number 5 implies all other numbers, because its meaning is rooted in its place in a sequence. And everything is like that — frank

Yes indeed.

all other numbers — frank

Hmmm.

All other natural numbers? Integers? Rationals? Reals? Complex numbers?

I think the answer is:

And every thing is like that. — frank

But Josh is the one who's actually read Husserl. Take it up with him probably.

Here's how to raise the issue of "further distinguishing" the SEP definition:

"There are abstract mathematical objects whose existence is independent of us" etc.

You simply ask, "What do you mean by 'existence'?" There is no one obvious reply. What are we supposed to say? -- "You know, exist, be. The opposite of not-exist. Case closed." Hopeless, and to make it worse, this so-called definition acts as if it is settling the matter, just by using the word and expecting readers to import their own concept of 'existence'. It looks like it is defining a certain kind of existence -- platonic existence -- but that's not possible without first knowing how existence itself is being construed.

Honestly, this isn't meant to be merely verbal gymnastics. I'm trying to demonstrate what I think is an important and all-pervasive issue, namely that there is no such thing as a sentence using 'exist' which can settle the question within the sentence itself of what 'exist' means. Again, this could be written using quantificational language, but I think it comes up often enough in ordinary discourse. A discussion of platonism is a great example. I'll leave Harry and Sally alone and just say: One person thinks there are abstracta which exist independently of us; another person says there are not. Why should we think they are both working with the same concept of what it means to exist? Indeed, if they were, wouldn't the issue be quickly resolved? The SEP talks about abstracta "whose existence is independent of us." Very well; what does SEP mean by 'existence'? Does it refer to a dimensional embodiment? being the subject of a proposition? being rationally apprehendable? being thinkable? being the value of a bound variable? something non-contingent? etc. etc. Thus the definition of mathematical platonism has told us absolutely nothing about what it means to exist. It cannot, formally.

What I do say is that material objects are perceived by the senses and so can’t be truly mind-independent, because sense data must be interpreted by the mind for any object to be cognised. — Wayfarer

Right you're saying the cognition of the objects is mind-dependent, and I have no argument with that since it is true by definition. But it doesn't seem to follow that the objects cognized are mind-dependent.

What interests me about the passage I quoted, is that mathematical functions and the like are not the product of your or my mind, but can only be grasped by a mind. — Wayfarer

Again I have no argument because it is only minds (and in a different sense hands and other implements) that grasp. It seems undeniable that a differentiated and diverse world is given to the senses, and that we experience that world in ways that are unique to the human, just as other animals presumably experience the world in ways unique to them. So, it seems to me that we are presented with number, that is numbers of things, and we abstract from that experience to conceptualize numbers.

The underlying argument is very simple - it is that number is real but not materially existent. And reason Platonism is so strongly resisted is because it is incompatible with materialism naturalism on those grounds, as per the passage from the Smithsonian article upthread, ‘What is Math?’: 'The idea of something existing “outside of space and time” makes empiricists nervous.' — Wayfarer

This is where I disagree; for me number is real and materially instantiated in the diversity of forms given to our perceptions. I don't "resist" platonsim, I simply don't find it plausible. It seems to me you try to dismiss disagreement with platonism by psychologizing it, by assuming it somehow frightens those who don't hold with it.

I think that is to greatly underestimate the intelligence and intellectual honesty of those you disagree with. I could do a similar thing by saying that people believe in platonism because they are afraid to admit and face the fact that this life is all there is. But I don't say that because I respect different opinions, and because dismissing arguments and worldviews on psychological grounds is shallow thinking. I don't claim that all platonists are stupid or afraid.

A very helpful explication, thanks for taking so much trouble with it.

I think my question gets addressed in this passage:

The first step of constitution of a multiplicity is the awareness of the temporal succession of parts, each of which we are made aware of as elements “separately and specifically noticed”. In the case of numbers, one must abstract away everything else about those elements (color, size, texture) other than that they have been individually noticed as an empty ‘unit’. — Joshs

This helps me imagine the process Husserl is speaking of, but I'm still left wondering what counts as a "part" or "element" (and these would presumably also be the "concrete contents" mentioned earlier). It is from these parts or elements that we must first abstract away qualities like color, size, and texture, and then engage the remainder -- the empty "unit" -- in the multiplicity-constituting process.

So . . . can this process take place with any physical series? Would Husserl countenance using an apple, say, as the starting part or element? Does it matter where we start? I think the answer is, "Sure, anything at all will do, as long as its perception counts as a 'sense act'," but I want to get your take on it.

So . . . can this process take place with any physical series? Would Husserl countenance using an apple, say, as the starting part or element? Does it matter where we start? I think the answer is, "Sure, anything at all will do, as long as its perception counts as a 'sense act'," but I want to get your take on it. — J

We have already indicated the concreta on which the abstracting activity is based. They are totalities of determinate objects. We now add: "completely arbitrary" objects. For the formation of concrete totalities there actually are no restrictions at all with respect to the particular contents to be embraced. Any imaginable object, whether physical or psychical, abstract or concrete, whether given through sensation or phantasy, can be united with any and arbitrarily many others to form a totality, and accordingly can also be counted. For example, certain trees, the Sun, the Moon, Earth and Mars; or a feeling, an angel, the Moon, and Italy, etc. In these examples we can always speak of a totality, a multiplicity, and of a determinate number. The nature of the particular contents therefore makes no difference at all. This fact, as rudimentary as it is incontestable, already rules out a certain class of views concerning the origination of the number concepts: namely, the ones which restrict those concepts to special content domains, e.g., that of physical contents.
(Philosophy of Arithmetic)

Good. Interesting that he includes psychical, abstract, and imaginary objects. I would have thought that contradicted the idea of "concrete data," but God knows how the German reads. The point is clear, in any case.

And I just noticed that "sense act" probably doesn't refer to what, in English, we mean by "sense perception," but rather to an "act of making sense."

number is real and materially instantiated in the diversity of forms given to our perceptions. — Janus

The nature of the particular contents therefore makes no difference at all. This fact, as rudimentary as it is incontestable, already rules out a certain class of views concerning the origination of the number concepts: namely, the ones which restrict those concepts to special content domains, e.g., that of physical contents.

I think that is to greatly underestimate the intelligence and intellectual honesty of those you disagree with — Janus

I’m not criticizing individuals but ideas. In this case, empiricist philosophy which can’t admit the reality of number because of it being ‘outside time and space’. If you take that as any kind of ad hom, it’s on you.

Where else would we have gotten the concept of number other than from the things around us?

I’m not criticizing individuals but ideas. In this case, empiricist philosophy which can’t admit the reality of number because of it being ‘outside time and space’. If you take that as any kind of ad hom, it’s on you. — Wayfarer

You said "Makes empiricists nervous". Empiricist philosophy can consistently admit the reality of number as instantiated in the things we encounter every day. You know, for example, like ten fingers and ten toes...

Every sentient creature is surrounded by objects but only rational sentient beings know arithmetic. Anyway if you read the quote in context it makes a point which is clearly salient to the OP (although I’m not going to try and explain it all over again.)

That's completely irrelevant to the point that number is empirically instantiated

It’s a perfectly meaningless expression. But Happy Christmas, regardless. :party:

Are you denying that there are great numbers of empirical objects? Anyway, Happy Christmas regardless to you too.

I read the Tyler Burge paper. It gives a convincing case for viewing Frege as a pure mathematical platonist. I hadn't known that Frege used the term "third realm" in such a similar way to Popper.

The key difference between Frege and Popper here is, as both @Banno and @Janus allude to, whether the 3rd realm exists independently of human thought, or is created by our thought. If Burge is right, then there's no doubt what Frege believed: complete independence. Popper stakes out a middle ground. In Objective Knowledge, Popper says:

The idea of autonomy is central to my theory of the third world: although the third world is a human product, a human creation, it creates in its turn . . . its own domain of autonomy. — Objective Knowledge, 118

And in fact, he chooses natural numbers as his example for how this works:

The sequence of natural numbers is a human construction. But although we create this sequence, it creates its own autonomous problems in its turn. The distinction between odd and even numbers is not created by us; it is an unintended and unavoidable consequence of our creation. — Objective Knowledge, 118

This is odd (sorry!) at first, but Popper goes on to explain that there are "facts to discover" about our human 3rd-world products. I think his use of "unintended" is key to understanding what he means. Just because I have created or invented something, it doesn't mean that in the act of doing so, I find myself in complete command, or complete awareness, of every single fact about my creation. And this does seem plausible with regard to numbers. If the number series is indeed invented, pace Frege, it's easy enough to imagine that early users would then discover that certain numbers -- invented merely for counting purposes -- had the quality of being either odd or even. This was never intended, but is certainly a fact for all that. Same with multiples, and primes, and on and on.

It's even more intuitively clear with regard to products we tend to agree are human creations. When I write a piece of music, I am very far from "intending" everything the music contains. In the process of (hopefully) improving what I write, I absolutely do discover things that are really there, but that I was not aware of when I wrote the music. Often enough, the discoveries are unpleasant, and I have to revise accordingly. But sometimes I find connections or implications that are fruitful and aesthetically interesting; they feel like genuine, "autonomous" facts about the music. Yes, I created the whole thing, but no, that doesn't mean I understand it completely. Only God, one supposes, creates in that fashion.

So anyway, Popper demonstrates that we can believe in all sorts of abstracta without needing to be platonist about it, and also without giving up the sense of discovery that goes along with exploring the 3rd realm.

If anyone is spending their holiday on TPF, poor devils, then Merry Christmas!

If the number series is indeed invented, pace Frege, it's easy enough to imagine that early users would then discover that certain numbers -- invented merely for counting purposes -- had the quality of being either odd or even. — J

Yes.

But we can go further than Popper. This, and this, and that, all count as one of something. These, and those, as two. That's an intentional act on our part, which is not only concerned with the things in the world but also concerned with ways of talking about those things. We bring one and two into existence, by and intentional act - it's something we do. Some important aspects of this. First, its we who bring this about, collectively; this is not a private act nor something that is just going on in the mind of one individual. Hence there are right and wrong ways to count. Next, the existence had here is that of being the subject of a quantification, as in "Two is an even number". Notice that this is a second-order quantification: Supose we say that there are two marbles and two flowers. We have not thereby created a third thing within the domain of discourse. We still only have two marbles and two flowers. When we say that two is an even number, we are still talking about marbles and flowers.

Do infinitesimals exist (in the platonistic sense)? — Michael

Infinitesimals exist. They are a higher-order quantification that can itself be quantified. Adding "in the Platonic sense" serves only to confuse what is going on.

This s what I tried to explain here:

Michael's argument talks about the existence of sentences. Hence it make use of quantification in a second-order language - a language about language. In a first-order language we can make an inference by quantifying over a predication - from f(a) to ∃(x)f(x). In second order logic one might perform a similar operation over a group of predicates. If we have ϕ(f(a)), we can infer ∃Pϕ(P) - if f(a) is ϕ, then something (P, in this case) is ϕ. But at issue here is a choice in how this is to be understood. Is it about just the things (a,b,c...) that make up the domain of the logic, or does it bring something new, P, into the ontology? The first is the substitutional interpretation, the second is the quantificational interpretation. This second interpretation has Platonic overtones, since it seems to invoke the existence of a certain sort of abstract "thing". — Banno

A note on logic. Natural languages are free to range over any topic and to say all sorts of strange things. Logic allows us to tie down what we can say with some level of consistency and coherence. The relation between higher-order logics described here sets out a way of talking about concepts without giving them some mystical "platonicistic sense". That's why the logic is useful, as a guide to language use, not as a replacement for natural languages.

Plato's approach was too muddled to be useful. Higher-order logic and intentionality provide a much clearer picture without the mysticism. It explains how such things as numbers can be said to exist when they are clearly not like chairs and rocks. It explains why mathematicians feel like they are 'discovering' things - they are. It gives precision to

I am inclined to argue that maths do not 'exist' in any objective sense. — Tzeentch

How do you apply that to these examples of the Fibonacci sequence? — frank

This is a good question. What the Fibonacci sequence gives us is a way of talking about the things you picture. It doesn't provide an explanation of why the shell follows that sequence. But it's not hard to find one.

All the snail does is to add calcium to the edge of it's shell. Each new shell chamber it grows is built on the previous two shell chambers. If we say the first is size one, then the second is grown on that, and is also size one. The third will be grown on those last two, and so be size two. The fourth is grown on the previous two, and so is size three.... 1,1,2,3,5... and so on.

Think of these as cross-sections of each chamber.

Snails do not have access to a platonic reality. It's not some mystical or divine intervention, but a simple result of a snail adding calcium to the edge of it's shell.


But we have a language that can talk about this growth.



Edt: Here's more than you ever needed to know about mollusc shells:

The key difference between Frege and Popper here is...whether the 3rd realm exists independently of human thought, or is created by our thought. If Burge is right, then there's no doubt what Frege believed: complete independence. Popper stakes out a middle ground — J

Compare:

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets." ' Furthermore in The Basic Laws of Arithmetic he says that 'the laws of truth are authoritative because of their timelessness: they "are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this, that they authority for our thought if it would attain to truth." — Tyler Burge

Intelligible objects must be independent of particular minds because they are common to all who think. In coming to grasp them, an individual mind does not alter them in any way, it cannot convert them into its exclusive possessions or transform them into parts of itself. Moreover, the mind discovers them rather than forming or constructing them, and its grasp of them can be more or less adequate. Augustine concludes from these observations that intelligible objects must exist independently of individual human minds. — Cambridge Companion to Augustine

Plainly Augustine has theological commitments that Frege lacks, but nevertheless the Platonist elements they have in common are significant. Augustine adds that reason is: “a kind of head or eye of our soul ... which does not belong to the nature of animals” (lib. arb. 2.6.13).11", clearly a reference to the tripartite soul of Plato, in which reason is a governing faculty, responsible for wisdom and seeking truth. Frege's notion that logical laws are "boundary stones set in an eternal foundation" parallels Plato's Forms and Augustine's intelligible objects as timeless, immutable realities. They are not dependent on human minds, cultures, or contingent physical realities but are 'discernable by reason', where 'reason' represents the faculty that is capable of grasping incorporeal truths.

We bring one and two into existence, by and intentional act - it's something we do. — Banno

Hence, these MUST be understood as constructions, hence contingent facts, our own creations, in fact, not immutable truths, which still retain a theological undertone that does not sit well with our secular age. Thomas Nagel quotes C S Peirce:

The only end of science, as such, is to learn the lesson that the universe has to teach it. In Induction it simply surrenders itself to the force of facts. But it finds . . . that this is not enough. It is driven in desperation to call upon its inward sympathy with nature, its instinct for aid, just as we find Galileo at the dawn of modern science making his appeal to il lume naturale. . . . The value of Facts to it, lies only in this, that they belong to Nature; and nature is something great, and beautiful, and sacred, and eternal, and real - the object of its worship and its aspiration.

The soul's deeper parts can only be reached through its surface. In this way the eternal forms, that mathematics and philosophy and the other sciences make us acquainted with will, by slow percolation, gradually reach the very core of one's being, and will come to influence our lives; and this they will do, not because they involve truths of merely vital importance, but because they [are] ideal and eternal verities. — Evolutionary Naturalism and the Fear of Religion

This is part of the preamble in which Nagel then describes the 'fear of religion' as one of the main motivations for the rejection of Platonism and the adoption of evolutionary naturalism:

Even without God, the idea of a natural sympathy between the deepest truths of nature and the deepest layers of the human mind, which can be exploited to allow gradual development of a truer and truer conception of reality, makes us more at home in the universe than is secularly comfortable.

That's the cultural dynamic that I think is behind the rejection of platonism in mathematics and the subsequent relativisation of reason.

If anyone is spending their holiday on TPF, poor devils, then Merry Christmas! — J

Beats crossword puzzles! And, same to you. :party:

these MUST be understood as constructions, hence contingent facts, our own creations, — Wayfarer

Rather, these CAN be understood as constructs. If you feel you need to include, in addition, a god or a platonic realm or whatever, then that's your choice.

a platonic realm — Banno

What do you think that might comprise? An ethereal palace, replete with ideal dogs and cats?

I've no idea what a Platonic realm might comprise— why would you ask me? Indeed, why ask at all, since the notion of a Platonic realm is fantasy.

Oh, so you don't know what it is, but you do know it's a fantasy.

You can think that, if it suits your narrative.

On a more serious note, there's an excellent current text available online which provides a succint and accurate account of the Platonic forms - Eric S Perl, Thinking Being - Introduction to Metaphysics in the Classical Tradition (.pdf). The chapter on Reading Plato. As it is directly relevant to the OP, I'll quote one passage at length.

Is there such a thing as health? Of course there is. Can you see it? Of course not. This does not mean that the forms are occult entities floating ‘somewhere else’ in ‘another world,’ a ‘Platonic heaven.’ It simply says that the intelligible identities which are the reality, the whatness, of things (such as "health") are not themselves physical things to be perceived by the senses, but must be grasped by thought.

It is in this sense, too, that Plato’s references to the forms as ‘patterns’ or ‘paradigms’, of which instances are ‘images,’ must be understood. All too often, ‘paradigm’ is taken to mean ‘model to be copied.’ The following has been offered as an example of this meaning of παράδειγμα (parádeigma) in classical Greek: “[T]he architect of a temple requiring, say, twenty-four Corinthian capitals would have one made to his own specifications, then instruct his masons to produce twenty-three more just like it.” Such a model is itself one of the instances: when we have the original and the twenty-three copies, we have twenty-four capitals of the same kind. It is the interpretation of forms as paradigms in this sense that leads to the ‘third man argument’ by regarding the form as another instance and the remaining instances as ‘copies’ of the form. This interpretation of Plato’s ‘paradigmatism’ reflects a pictorial imagination of the forms as, so to speak, higher-order sensibles located in ‘another world,’ rather than as the very intelligible identities, the whatnesses, of sensible things.

But forms cannot be paradigms in this sense. Just as the intelligible ‘look’ that is common to many things of the same kind, a form, as we have seen, is not an additional thing of that kind. Likewise, it makes no sense to say that a body, a physical, sensible thing, is a copy, in the sense of a replica or duplicate, of an intelligible idea. Indeed, Plato expressly distinguishes between a copy and an image: “Would there be two things, that is, Cratylus and an image of Cratylus, if some God copied not only your color and shape, as painters do, but also … all the things you have? — Eric D Perl Thinking Being, p31 ff

I say that 'forms' are much more like 'intelligible principles' than what they are often confused for, which is a kind of ethereal shape. I think much of the dismissal of them is based on centuries of poor schoolroom teaching by those who really hadn't grasped that fact. But there are contemporary sources, such as Rebecca Goldstein's Plato at the Googleplex, and Iris Murdoch's Sovereignty of the Good, which provide a much more nuanced account of their continuing relevance.

Sure, all that, there might be a more accomodating reading of what I said above that can be made compatible with Platonic texts. The difference is that, at least to my eye, the account I gave above indicates how stuff like numbers and property and so on are constructed, by modelling that construction in a higher order logic.

It's not a novel account I believe.

I'm not overly interested in defences of Plato or Thomisim, or even Popper or Searle. Thier value is in what they help us understand.