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!