On Logic and Mathematics

Quoting Pfhorrest

simple spelling or grammar

is a direct applications of my

for which were are simply not

without concerning itself with what anyone might be communicating about what [about which of the various possible?] attitudes toward them.

My struggle with that one may be due more to my philosophical prejudice against attitudes than to any capital problem for the sentence. Can't quite tell.

And so on with all those we can replicate [:?] implication,...

whether there are, or or whether

reality being describable by a formal language would be either that ome

, continuous with the one we find ourselves in and the of same nature as it;

the question of whether were are

Interesting and worthwhile. Thanks for sharing.

Reply to bongo fury Thank you for the proofreading! I'll put these on my to-do list and will fix them as soon as I get the chance.

Reply to bongo fury Those fixes are all implemented now.

@fdrake I would love your input on this one. I feel like I'm way out on a limb outside my area of expertise talking about all of this mathematics stuff, and your contributions to my Mathematicist Genesis thread (which was inspired by writing this essay) show that you have a much deeper understanding of the topic than I do, so I'd love to know if I got any of the math horribly wrong in this.

@Wayfarer It strikes me that the very end of this might be of interest to you, if you haven't read it already.

Reply to Pfhorrest It looks an interesting paper! But my view on this matter is actually simple one: that logical laws, real numbers, and the like, are real but not material. That's all there is to it [sup] 1[/sup]. When we grasp an arithmetical truth or logical principle or concept, we're grasping something that is not simply 'in our minds' but is real regardless of whether we grasp it or not. In other words, it is real independent of any act of thought. But it can only be grasped by reason: it is an 'intelligible object', that is, an object of pure thought. I understand this principle as something fundamental about 'objective idealism' which in my humble opinion is really the mainstream of Western philosophy.

In fact, I've discovered that this is a vital strain running from Pythagoras through Plato through Augustine to the Western philosophical tradition, generally. Even to this day there are mathematical Platonists, such as Kurt Godel

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

So, you see, this opens up the perspective that there are "real abstractions"; they're not simply 'in the mind'. That designation is the convenient way that modern physicalism sweeps this problem under the rug: they're simply 'in the mind' therefore 'products of the brain' and can be understood through the prisms of biological adaptation and neurology. But this gets the whole thing backwards, because logical laws, real numbers, and so on, in some sense must always be so - that is why their apprehension is designated 'a priori'. Certainly, we evolved to the point where we can recognise them, but that doesn't make them explicable in evolutionary terms.

The domain of logical relations has been in other times and places designated as the 'formal realm' - which is the domain of laws, numbers, logical (as distinct from material) necessity. But all this has little bearing on your essay! (Nevertheless, have a brief peruse of this passage from the Cambridge Companion to Augustine on the topic of 'intelligible objects'.)

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1. And the reason that is controversial, is because of the mainstream view that only material things are real.

A couple of quick comments:

(1) Your theory of mood is very similar to the way most formal semanticists treat mood. Already Lewis in his "General Semantics" (cf. pp. 220ff of his Philosophical Papers) he proposed to analyze sentences as containing a sentence radical and a mood operator. That is, however, the easy part of the analysis. The difficult part is to specify a formal semantics for these syntactic operators. One idea, close to possible world semantics, would be to treat proposition radicals as sets of possible worlds and then treat the declarative as assigning the same function as the propositional radical (thought of as its characteristic function), the interrogative as assigning partitions of possible worlds to the proposition radical, and similar to the other moods. An up-to-date treatment of moods along these lines is given by Paul Portner's Mood (Oxford University Press, 2018), which I highly recommend.

(2) On Platonism, I don't know any current platonists (with lower case "p") who defend that concrete objects "partake" in abstract objects. Most platonists, in fact, think that there is no interaction whatsoever between abstract and concrete objects, since any such interaction would have to take place at least in time, and abstract objects are (generally, though not always) thought to be outside space-time. There are a host of problems here with "mixed" objects (e.g. {Nagase}, the singleton whose only member is me: is that abstract? concrete? partially abstract and partially concrete? is it abstract with concrete parts?), but ignoring those, I think most platonists would be happy to think that there is a complete separation between the two types of objects. Their commitment to abstract objects is wholly theoretical: our best theories need abstract objects in order to be true, hence, those objects exist.

(3) Finally, it is not clear to me how your proposed solution works. You're proposing that mathematical objects are actually concrete? So they inhabit space-time (such that I could kick the number 2, for instance)? Or are you proposing that nothing is concrete?

Incidentally, for what is worth, I would highly object to treating logic as dealing with relations between "ideas" or laws of thought, or whatever. This is completely misses the point of what logicians such as Saharon Shelah are doing (I'm thinking of his classification theory). Of course, some people are happy to bite the bullet and simply classify his work as some kind of more abstract algebra, but I personally think that there is some philosophical payoff to treat his program as being engaged with logic.

Reply to Nagase Thanks for the reference to Paul Portner, I'll have to look him up when I can.

Can you elaborate more on the alternatives to "partaking" supported by contemporary platonists? I didn't mean to imply that platonists support any kind of spatiotemporal interaction between abstract and concrete objects, but certainly on a platonist account the abstract Form of the Triangle is somehow present or involved or [something] in some way in an actual concrete triangular object, no? What phrasing would a contemporary platonist use to describe that relationship(?), instead of "partakes"?

Quoting Nagase

(3) Finally, it is not clear to me how your proposed solution works. You're proposing that mathematical objects are actually concrete? So they inhabit space-time (such that I could kick the number 2, for instance)? Or are you proposing that nothing is concrete?

Neither really, but closer to the latter. I was actually about to reply to @Wayfarer asking if he understood this part of my position when I saw your reply, so I'll let this serve as a reply to both. I'm proposing that there is not a hard ontological difference between abstract and concrete objects, that everything is ontologically like how abstract objects are usually reckoned to be, and the concrete world (with its space and time, all of the concrete objects it contains, including ourselves) is an "abstract" object, differing only from other "abstract" objects in that it is the one of which we are a part; so truly "abstract" objects are just objects like our concrete world, of which we are not a part. I see this as still compatible with physicalism in that any sentient beings that exist as part of the structure of any other "abstract object" will find that object to be a physical world that they inhabit, just as we find ours. (And they will find our world as accessible only to the intellect, i.e. only something they can imagine, as we find theirs).

Just as, in modal realism, other possible worlds are ontologically the same as our actual world, and anyone who exists in another world that is to us merely possible but not actual would find that world actual to themselves. So it's not really accurate to say that all possible worlds are actual, or that nothing is actual. And likewise, it's not really accurate of my view to say that mathematical objects are concrete, or that nothing is concrete. "Concrete is indexical", as I said in the essay.

(Obviously, not every abstract object has a structure that could include sentient observers within it; nobody's going to find themselves existing as a part of the abstract form of the triangle. But likewise nobody's going to be existing in a possible world where as much antimatter as matter was produced in the big bang, but that doesn't make such possible worlds ontologically different from others).

Lastly, for Wayfarer, the part I thought he might find most interesting: I hold that a lot of what we think of as "concrete objects" are really abstractions away from the most fundamentally concrete (to us) reality, the occasions of our experience. Rocks and trees and tables and chairs are abstractions from those occasions of experience, but still grounded in that experience and so still "partially concrete". We're projecting the existence of abstract objects "behind" the experience, to structure and make sense of it, in a way similar to the noumenal realm projected to exist "behind" the phenomenal realm on Kant's account: we can't actually have any true experience of those abstract things in their true selves, we can at best guess at that from the concrete experiences that we have. "Fully abstract" objects are those completely divorced from experience, and just found through imagining them.

Quoting Nagase

Incidentally, for what is worth, I would highly object to treating logic as dealing with relations between "ideas" or laws of thought, or whatever. This is completely misses the point of what logicians such as Saharon Shelah are doing (I'm thinking of his classification theory). Of course, some people are happy to bite the bullet and simply classify his work as some kind of more abstract algebra, but I personally think that there is some philosophical payoff to treat his program as being engaged with logic.

Can you elaborate more on this, because I don't understand what the objection is.

Quoting Nagase

Most platonists, in fact, think that there is no interaction whatsoever between abstract and concrete objects, since any such interaction would have to take place at least in time, and abstract objects are (generally, though not always) thought to be outside space-time.

This is a post-Cartesian way of thinking. It treats abstracts as 'objects', like regular objects, but in another domain. Then it wonders how 'abstract objects', being of a radically different kind, can influence concrete objects; which is exactly the problem of the 'ghost in the machine', and the problem of accounting for how an immaterial mind can affect a material body. These problems arise from Descartes' notion that res cogitans is something objectively existent. (A lot of this was the subject of Husserl's critique of Descartes in Crisis of the European Sciences.)

But what if abstracts are not objectively real; that's there's a sense in which they can be said to be real, but not necessarily existent.

This is discussed by Russell in Problems of Philosophy, On Universals. Note the bolded passage in particular.

Consider such a proposition as 'Edinburgh is north of London'. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth's surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe. This is, of course, denied by many philosophers, either for Berkeley's reasons or for Kant's. But we have already considered these reasons, and decided that they are inadequate. We may therefore now assume it to be true that nothing mental is presupposed in the fact that Edinburgh is north of London. But this fact involves the relation 'north of', which is a universal; and it would be impossible for the whole fact to involve nothing mental if the relation 'north of', which is a constituent part of the fact, did involve anything mental. Hence we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.

I think the other key phrase is this: 'we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.'

So I am arguing that real numbers and logical laws and the like are all of this nature; they are constitutive of judgement, and ipso facto of reality, for rational, language-using beings such as ourselves; but they don't exist 'out there somewhere'. The difficulty is precisely that they don't exist 'in the same sense in which London and Edinburgh exist'. This distinction introduces an equivocation into the word 'existence'; whereas the accepted view is that 'existence' has one meaning, that something either exists, or it doesn't. If we consider that number, logical laws and the like, exist in a different way than do concrete objects and particulars, then this is the key to understanding the nature of abstracts.

Reply to Pfhorrest

I don't think most platonists would recognize the need for any relationship between actual, concrete objects and abstract objects, at least not in this direct way. It is useful, however, to distinguish between two types of platonists here (I take this classification from Sam Cowling's Abstract Entities, a very useful---and opinionated---survey of the terrain). Expansive platonists include among the existing abstract objects not only numbers, pure sets, and propositions, but also what has commonly been called abstract types, such as, perhaps, musical works, poems, recipes, letters (in the sense of "a", "b", etc.), etc. Austere platonists countenance only numbers, pure sets, and, perhaps, propositions and properties. So expansive platonists may want a relation like "partake" between types and tokens, perhaps in the form of instantiation. But austere platonists will not need any such a relation. If there is any relation between concrete "triangles" (I use scare quotes to indicate that I'm not taking such "triangles" to be literally triangles, i.e. instances of an abstract type) and abstract ones, it will probably be one of approximation (say, we can map an abstract triangle up to a margin of error to the space-time points occupied by the concrete triangle), where this relation is itself an abstract set.

As for your proposal, I think there are a couple of separate issues here. First, there is the question of whether there is a conceptual distinction to be made between abstract and concrete objects. Notice that this is a conceptual question: perhaps there is such a distinction, but all objects fall only on one side of the divide (that is, e.g., the nominalist position), or perhaps the distinction is not exclusive, i.e. there are hybrids (which is what I mentioned with the problem of classifying {Nagase}). Still, it must be possible to make such a distinction independently of any ontological questions.

Second, there is the question of whether the ultimate constituents of reality fall on one side or the other of the division. Some people believe that reality is hierarchically structured, with metaphysical atoms at the bottom and every other object constructed out of such atoms (and constructs of such atoms) by way of metaphysical operations (perhaps "building" operations in the sense of Karen Bennett)---one model for this is a cumulative set-theoretical hierarchy with (or without) ur-elements, with the ur-elements (or the empty set) being the atoms and the "set-builder" operation being used to construct the other sets. Given this, it's possible to ask what reality is really like, that is, how to characterize the objects at the bottom (the fundamental objects, those that really, really exist): is it abstract or concrete?

Third, supposing the hierarchical structure picture is true, there is the question of the character of the constructed entities. Are they abstract, concrete, or hybrids? Notice that, if supposing there are hybrids, depending on the character of the building operations that you use in "constructing the universe", there may be a way to reduce everything to abstract entities. One influential program tried to do precisely this, supposing that everything could be constructed out of time-slices of atoms and sets of such time-slices (I think Quine held something like this). Here's the idea: pick any set of ur-elements; presumably, they are countable (if not, you'll need some choice to make the idea work). So map the ur-elements into the natural numbers and use this map to construct a set that is the extended image of the original set (more formally: if f is the original map, build f* recursively by setting f*(x)=f(x) if x is an ur-element, otherwise set f*(x)={f*(y) : y in x}). This will build an abstract replica of the original universe, preserving its structure. So everything you wanted to do with the original universe, you can do with the new universe. Simplicity considerations then may dictate that the new universe is the real universe. (I don't endorse this line of thinking, but it may be of interest to your program.)

Finally, about Shelah, my point is this: (A) some people define logic as xyz; (B) but xyz doesn't contemplate what logicians such as Shelah are doing; (C) therefore, xyz is not a good definition. I think this is true when you take "xyz" to be "the study of logical consequence relations", and doubly true when you take "xyz" to be "the study of relations between ideas". Personally, I'm attracted to a definition of logic as being the study of local invariant relations, but there is much here to work out...

Reply to Wayfarer

As I said in my reply above, I don't think platonists need to be saddled with such Cartesianism. There would be such a need if they thought there is any interaction between concrete and abstract objects, and wondered about the (per force, mysterious) character of such interaction. But, as I said, I don't think platonists are committed to there being any such interaction, and in fact I would argue that most (all?) deny that there is such an interaction.

Reply to Nagase I thought I had addressed those points, but apparently I have not succeeded.

Quoting Nagase

First, there is the question of whether there is a conceptual distinction to be made between abstract and concrete objects

...

Second, there is the question of whether the ultimate constituents of reality fall on one side or the other of the division

Yes, but the tricky bit here is, I think that the nature of that conceptual distinction is not what it is usually taken to be, and that is the distinguishing feature of my view, not the answer to either of these questions.

Tell me what you take the answers to these questions to be regarding Lewis' kind of modal realism, because I think the situation there is perfectly analogous. Does Lewis take there to be a conceptual distinction between actual and merely possible worlds? Does he take all worlds to fall on one or the other side of that division?

I think that Lewis would answer "yes" and "no" respectively, but that his opponents would answer likewise, and the disagreement between them is about what that distinction is like, not the answers to those two questions.

Lewis would say there is a distinction between actual and merely possible, that distinction being merely indexical, like the difference between "here" and "there" (there's nothing ontologically different about two different places, but "here vs there" still makes conceptual sense); and he would say that for any person there is one world that is actual to them and the rest are merely possible.

His opponents would agree that there is a distinction between the actual and merely possible, but that that distinction is ontological, that merely possible worlds have a fundamentally different ontological status than the actual world; and they would also agree that there is only one actual world, and the rest are merely possible.

My position regarding concrete vs abstract is perfectly analogous to Lewis' regarding actual vs merely possible.

Plato wrote that "we must agree that that which keeps its own form unchangingly, which has not been brought into being and is not destroyed, which neither receives into itself anything else from anywhere else, nor itself enters into anything else anywhere, is one thing. It is invisible". I think he marveled at the variety of these Forms (which were really pictures in his mind's eye) but thought them united somehow in a meta-Form. He did not think this was God it appears. Personalities were separate. Whether you think truth is the Prime Mover, or Jesus, or an impersonal Form (even if you call it Brahma), you are making truth into something you want to have a relationship with. Even if all that exists is the world, the truth that all that exists is the world is true but that truth rests on nothing. I think this is key to dealing with logic and it's subdivisions

Reply to Pfhorrest

I don't think the positions are analogous at all. Lewis can say that other worlds are real because he is assuming that to be a real world is to be a concrete entity (say, the mereological sum of all spatio-temporally connected parts of its domain). In your case, you're saying that abstract entities are real because... what? They are concrete? But then they are not abstract. That is, either there is a space-time in which a thing inhabits or there is not. If there is, then the thing is concrete, if there isn't, it is abstract. I don't see any way to relativize this distinction further.

Quoting Nagase

I don't see any way to relativize this distinction further.

I like how you phrased that. Great question

Quoting Nagase

I don't think the positions are analogous at all.

Being the originator of one of the two positions in question, I've defined it by analogy to Lewis's, so saying they're not really analogous is saying I'm not taking the position I say I am. But I think you're just not understanding the position I am stating at all somehow, so let's abandon that approach since it's obviously not working.

Here is a somewhat poetic way of putting it. If the platonist believes that there's the concrete material world and then "Plato's heaven" in which the abstract objects exist, and the nominalist says there is no such "heaven", just the concrete material world, I am saying that the concrete material world is an object in Plato's heaven. There isn't any space or time inherent in that "heaven", in which the abstract objects of it are arranged, but space and time are features of some abstract objects, like whatever abstract object is a perfect model of the physical world we experience, which just is our physical world, of which we are parts.

(The way this is analogous to Lewis's modal realism is that Lewis says there is no special property of "actualness" that ontologically differentiates the actual world from other merely possible worlds; the actual world is just one instance of the kind of thing that merely possible worlds are, which is only special because of its relationship to us. Likewise, the concrete world, on my account, isn't ontologically different from any abstract objects, it's only special because it's the abstract object of which we are parts).

Reply to Pfhorrest

Here is why I think the analogy is poor: for Lewis, "actual" is an indexical, because it is short for "in this space-time continuum". So, from the point of view of this space-time continuum, we're actual; from the point of view of another space-time continuum, they're actual and we're possible. So whether or not something is actual or possible depends on which space-time continuum you're in. With abstract and concrete, however, there is no such relativity: either you are in a space-time continuum, or not. If yes, then you're concrete; otherwise, you're abstract.

So, in your proposal, it's not like I'm abstract from one point of view and concrete from another. Rather, everything is abstract (everything is in Plato's heaven, to use your terminology), and we simulate concreteness by appealing to certain properties of abstracta. As I said, this is very similar to what (I remember) Quine proposed: take the pure sets, find the reals inside them, form [math]\mathbb{R}^4[/math], and identify the concrete objects with sets of points in this space-time surrogate. Of course, you can also go on and say that this simulation is what people have "meant" all along by concreteness, or perhaps simply follow Quine and say that, if this is not what they meant, so much the worse for them ("explication is elimination"). I don't personally find this Quinean route very appealing, but it not all that implausible...

Quoting Nagase

for Lewis, "actual" is an indexical, because it is short for "in this space-time continuum". So, from the point of view of this space-time continuum, we're actual; from the point of view of another space-time continuum, they're actual and we're possible.

Yes, and for me, "concrete" is an indexical, because it's short for "part of this mathematical structure". So, from the point of view of beings that are part of the mathematical structure that is the world as we know it, other things that are parts of that same structure are concrete, and other structures entirely are abstract. But from the point of view of beings that are part of those other structures instead, our entire world is abstract, and the other parts of their structure are the concrete things.

Quoting Nagase

So, in your proposal, it's not like I'm abstract from one point of view and concrete from another.

That is exactly what my proposal is saying.

Quoting Nagase

Of course, you can also go on and say that this simulation is what people have "meant" all along by concreteness, or perhaps simply follow Quine and say that, if this is not what they meant, so much the worse for them ("explication is elimination").

Yes, that's pretty much my approach.

The difficulty communicating this is, like I said before, the difference between my view and its negation is about what constitutes the difference between abstract and concrete, just like the difference between Lewis' view and its negation is about what constitutes the difference between merely possible and actual, so on the one hand each of us would kind of like to say things like (in Lewis' case) "the actual world is just one of many merely possible worlds, and all possible worlds are actual to beings who are part of them" or (in my case) "the concrete world is just one of many abstract objects, and all abstract objects are concrete worlds to beings who are part of them". But in both of those cases we'd be mixing up the senses of both terms, using them once in the sense of our opponents and once in our own sense.

Let's rephrase Lewis's position like this by tabooing his opponent's senses of "actual" and "possible" and replacing them with "AC" and "PO". Lewis' opponents say that there are two different ontological kinds of things, AC worlds (of which there is only one, the actual world) and PO worlds (of which there are many, all the other merely possible worlds). In contrast, Lewis says that there is just one kind of ontological thing, AC and PO are ontologically the same, call it "ACPO"; and the actual world is just this ACPO world, while other merely possible worlds are just other ACPO worlds.

Now rephrase my position by likewise tabooing the platonist/nominalist senses of "concrete" and "abstract", and replacing them with "CO" and "AB". The platonists and nominalists say that there are two different ontological kinds of things in concept, CO objects and AB objects, and they argue among each other about whether or not there exist any AB objects. In contrast, I say that there is just one kind of ontological thing, CO and AB are ontologically the same, call it "COAB"; and concrete objects are just parts of this COAB object, while abstract objects are just other COAB objects.

Reply to Pfhorrest

In that case, I don't have much more to add.

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Reply to tim wood

Thanks for the compliment (I think it was a compliment?)!

As for Kant, I'm not sure I understand your point. Yes, for Kant, space and time are imposed by our productive imagination onto the phenomena. But I don't see how this relates to the concrete/abstract distinction---unless you're saying that things, as considered in themselves, are neither concrete nor abstract? Is this your suggestion?

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Reply to tim wood

I'm not following (perhaps you're already regretting your statement?). Let us suppose, with Kant, that space and time are the form of our intuition, and are therefore the result of our productive imagination, i.e. ens imaginaria, as he puts it. Let us also grant that this means that things, as considered in themselves, are not represented as in space and time. How does this impact the abstract/concrete division? One can still hold that concrete refers to things in space-time, and that abstract, if it refers at all, refers to things not in space-time. To be sure, this would make things, as considered in themselves, to be an abstraction, but I don't think that is very far from what Kant was thinking.

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Quoting Nagase

space and time are the form of our intuition, and are therefore the result of our productive imagination,

Form of intuition, ok, but how does “...are therefore the result of our productive imagination...” follow from it?

The isolated reason can adopt three positions.

1) 2+2 equals everything but four, so there are no numbers

2) there is only one number

3) regular mathematics

Which of the 3 apply to the world is a question too

Quoting tim wood

A matter of language.

Boy howdy. That damn “in-itself”.....talk about a rabbit hole. So fundamentally necessary in the groundwork, so commonly mistaken in the dialogue. I concur with your objection.

Reply to Mww Reply to tim wood

I don't mind discussing Kant's theory of intuition or his conception of things in themselves, but I don't want to hijack this thread (specially since I want to go back to some of the things in the OP). Isn't it better to start a new thread?

Reply to Nagase

I hear ya.

I read everything but little interests me. Yours caused me to question my own interpretations, which is fine, Kant being the decent challenge he is.

Anyway...minor point en passant.

FWIW I do discuss these Kantian "categories" (or something much like them at least) in the next essay in this series On Ontology, Being, and the Objects of Reality.