The very idea of a psychological quality is already too substantial sounding to my ear. — apokrisis

Peirce acknowledged this - as soon as we talk or even think about a color or other quality, it is no longer 1ns in itself.

I've had some exposure to category theory. Not much but enough to know that modern math is done very differently than anything you see as an undergrad math major. Equations are replaced by arrow diagrams. It's a very different point of view. — fishfry

My only exposure to category theory (so far) is Zalamea's paper, which I am in the process of rereading because I suspect that it will make even more sense to me now than it did a few months ago. Peirce was a strong advocate of diagrammatic reasoning, but he did not confine the term "diagram" to visual representations; rather, a diagram is any sign that embodies the significant relations among the parts of its object. An algebraic equation is a diagram in this sense, but it is not as "iconic" as a geometrical sketch. This is precisely why Peirce developed the Existential Graphs, a diagrammatic system of logic whose three versions are equivalent to standard propositional logic (Alpha), first-order predicate logic (Beta), and certain kinds of modal logic (Gamma). He hoped that diagrammatic reasoning would be a means by which mathematics could overcome the limitations of the discrete and better account for true continuity.

I've always understood category theory to be loosely related to structuralism. We no longer care what things are made of, we care about their relationships to other things; and about very general patterns in those relationships that tie together previously unrelated areas of math. — fishfry

Several commentators on Peirce have suggested that his philosophy of mathematics was very similar to modern structuralism; e.g., this paper by Christopher Hookway, and this one by Paniel Reyes Cardenas.

Why have I never heard of Peirce before? — fishfry

Ah, the perennial question whenever someone discovers him for the first time. My reaction was exactly the same. Best I can tell, the key factors are:

• He never wrote any books compiling his insights, just lots of articles and tens of thousands of pages of unpublished manuscripts.
• His only academic position was as a part-time lecturer on logic at Johns Hopkins for a few years, so he mostly toiled in relative obscurity.
• Both of the above likely stem from his generally cantankerous demeanor and somewhat scandalous (for its time) personal life.

I still don't know what triadic Peircean phenomenology is. Can this be explained simply? — fishfry

Peirce believed that there are exactly three universal categories that are present in every phenomenon, and in order to avoid specific associations that might be too narrow, he preferred to call them simply Firstness (1ns), Secondness (2ns), and Thirdness (3ns). As I mentioned above, there are various ways to differentiate them - possibility/actuality/necessity, quality/fact/law, spontaneity/reaction/habit, feeling/action/thought. Logically, they correspond to monadic/dyadic/triadic relations; Peirce postulated, and Robert Burch proved (much later), that each of these is irreducible to the others, while all relations of tetradic or higher adicity can be reduced to triadic ones.

It seems to me that you're trying to argue against mathematical terminology without actually understanding the mathematics involved. — Michael

He might very well understand it, he just refuses to accept it. He is committed to the presupposition that only the actual is real, so if something is actually/nomologically impossible, it just is impossible, full stop.

When you assert that all natural numbers are countable, this is an inductive conclusion. — Metaphysician Undercover

No, it is a deductive conclusion that is necessarily true, given the standard mathematical/set-theoretic definition of countable/denumerable/enumerable/foozlable.

Cardinality is defined on Wikipedia as " a measure of the 'number of elements of the set'. It seems quite obvious that it is impossible to have a measurement of the number of elements in an infinite set. — Metaphysician Undercover

Not if cardinality/multitude is defined in a particular way that specifically pertains to infinite sets. For any set with N members, there is a "power set" that consists of all of its subsets, and that power set has 2^N members. For any value of N whatsoever, 2^N > N. Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. That is all it means to say that the set of real numbers is "bigger" than the set of natural numbers. I will grant that it is counterintuitive, but it is defined this way in set theory, where it is not problematic at all since mathematics is the science of drawing necessary conclusions about ideal states of affairs; it has nothing to do with actual states of affairs.

Being infinite, you cannot establish a bijection, just like you cannot count them. You might assume that if you could count them, you could place them in a one to one correspondence, but you cannot count them, so such an assumption is irrelevant. — Metaphysician Undercover

Being able to finish counting them has absolutely nothing to do with establishing a bijection, or one-to-one correspondence. @Michael's subsequent reply is incorrect - bijection applies to infinite sets, as well as finite sets; it is a specific type of surjection.

So that does conflict with some of Peirce's apparent definition of 1ns as brute quality (with its implications of already being concrete or substantial actuality). — apokrisis

No, I understand his 1ns in itself to be quality as possibility, or unembodied quality; medad rather than monadic predicate. Anything brute and/or actual is 2ns.

So continuity or synechism itself is 3ns - but 3ns that incorporates 2ns and 1ns within itself. — apokrisis

Agreed, 3ns involves 2ns and 1ns, and 2ns involves 1ns.

So 1ns (in a misleadingly pure and reified sense) is vagueness (a certain unconstrained bruteness of possibility - as in unbounded fluctuations). — apokrisis

Again, Peirce did not use "bruteness" to refer to 1ns, only 2ns.

So 3ns is necessity, 2ns is actuality, and 1ns is possibility. — apokrisis

That is certainly one manifestation of the categories. Others include quality/fact/law, spontaneity/reaction/habit, and feeling/action/thought. I also think that 3ns is often conditional necessity, rather than absolute necessity.

... see Menno Hulswit's excellent books and papers on this issue ... — apokrisis

Yes, I have read a bunch of his stuff, although it has been a while.

But once you get used to it, it all makes sense. — apokrisis

It definitely takes some getting used to, but is well worth the time and effort.

And I expect you already get most of this. But just in case, that is a summary of why the answer is not so straightforward. — apokrisis

Thanks, it definitely helps me map the terminology that you tend to use around here to Peirce's own.

And a better paper on the Peircean project is probably... http://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf — apokrisis

So at the moment I'm coming to Peirce by way of Zalamea's exposition of the mathematical philosophy of Grothendieck. — fishfry

I was delighted to learn this evening that Zalamea has agreed to a "slow read" of this very paper via the Peirce-L e-mail list in the near future. If you are interested in joining that conversation, or even just monitoring it, I can keep you posted on the details as they are worked out. He hopes to participate himself, although likely to only a limited extent, because he apparently intends to go into seclusion over the next couple of years in order to focus on writing a lengthy new monograph on Grothendieck.

In the beginning was the word, eh? — Wayfarer

The word plus the vagueness it could organise. — apokrisis

Peirce: "If we are to explain the universe, we must assume that there was in the beginning a state of things in which there was nothing, no reaction and no quality, no matter, no consciousness, no space and no time, but just nothing at all. Not determinately nothing. For that which is determinately not A supposes the being of A in some mode. Utter indetermination. But a symbol alone is indeterminate. Therefore, Nothing, the indeterminate of the absolute beginning, is a symbol. That is the way in which the beginning of things can alone be understood." (EP 2:322; c. 1904)

There was the vagueness that would be utterly patternless and directionless action. — apokrisis

Peirce: "In that state of absolute nility, in or out of time, that is, before or after the evolution of time, there must then have been a tohu-bohu of which nothing whatever affirmative or negative was true universally. There must have been, therefore, a little of everything conceivable." (CP 6.490; 1908)

Genesis 1:2: "The earth was without form and void (Hebrew tohu wa bohu) ..."

And someone said that's a little boring. Let's tweak it with some contrast. Let's add some constraints to give it some light and dark. Let's create a little story about differentiated being. — apokrisis

Someone? Now you are just teasing me.

Peirce, describing the author of Genesis 1:2-5: "His tohu bohu, terra inanis et vacua is the indeterminate germinal Nothing. His Spiritus Dei ferebatur super aquas is consciousness. His Lux is the world of quality. His fiat lux is an arbitrary reaction. His divisit lucem a tenebris is the recognition of the necessary duality. His vidit Deus lucem quod esset bona is the waking consciousness. Finally; his factumque est vespers et mane, dies unus is the emergence of Time." (NEM 4:138; c. 1898)

That is, either the judgement of "continuous" or "discrete" would be determinations imposed (counterfactually) on pure possibility. — apokrisis

In this context, do you basically see continuity as 3ns, discreteness as 2ns, and possibility as 1ns?

The "continuity" then is of the systematic "constraints plus freedoms" kind that I employ. — apokrisis

Given that existence is 2ns, do you generally prefer to characterize 3ns as "constraints" and 1ns as "freedoms"?

You can take my word for it that .999... = 1 is a theorem of nonstandard analysis. But actually you don't need to take my word, I provided a proof above. — fishfry

I have no reason to doubt that you are correct about this. Thanks for another helpful clarification, especially since @tom chose for some reason not to provide one.

You cannot create a new number by adding an infinitesimal quantity. — tom

I am neither a mathematician nor a philosopher, but that statement seems consistent with the claim that the real numbers do not qualify as a true continuum in Peirce's sense, since they skip over those infinitesimal intervals. If you disagree, I would sincerely appreciate an explanation why I am mistaken about this. Have you read the Zalamea paper, or at least its first chapter that spells out the essential properties of a Peircean continuum?

As it happens, .999... = 1 is a theorem even in nonstandard analysis. — fishfry

I will take your word for it, but my understanding is that the precise nature of the relationship between Peirce's continuum and nonstandard analysis is still not fully settled.

It's you. — tom

I doubt it, since I have been clearly saying all along that a true continuum is not made up of "individual points or their aggregates." I might even agree with that whole post, if I am understanding it correctly. In any case, we might as well wait for @SophistiCat's own answer.

Which real number is bigger? 1 or 0.999... — tom

We typically treat the two values as equal, but arguably there is an infinitesimal (non-zero) difference between them. As you might have guessed, the Peircean continuum is non-Archimedean.

Are your comments directed at any particular person or post?

Breaking sand "infinitely" yields tiny bits of sand? Please! — tom

It obviously does not yield an unbroken continuum.

Breaking the real numbers "infinitely" yields what? — tom

More real numbers, all of which are distinct; again, it obviously does not yield an unbroken continuum. Put another way, the set of real numbers has a multitude or cardinality, which is exceeded by its power set; but a true continuum exceeds all multitude or cardinality. It is not composed of parts, it can only be divided into parts, all of which can likewise be divided into more and smaller parts of the same kind.

Right - the real numbers constitute an analytic continuum, but not a synthetic continuum; i.e., a true continuum in the Peircean sense, which cannot be represented by numbers at all. As he put it, "Breaking grains of sand more and more [even infinitely] will only make the sand more broken. It will not weld the grains into unbroken continuity."

Our fundamental theories are based on the continuum of space-time. — tom

An analytic continuum (real numbers), or a true continuum (Peircean)?

I hope you noticed that computers can't even instantiate the reals. — tom

How is that significant in the context of this thread?

I am not suggesting that musical notes are not useful. What I am saying is that they, as are all symbols, are an absolute awful representation of the experience itself. — Rich

It is trivially true that no representation reproduces its object in every respect, and the purpose of musical notes - and mathematical symbols/equations - is obviously not to represent "the experience itself."

We must put mathematical equations in proper perspective and not get carried away by them. — Rich

Agreed, but we also should not undervalue them, either.

As I just finished practicing piano, I observed that the notes that I was playing as I read them off paper was can awful representation of the music that I was creating. — Rich

But without that representation, you would not be able to play that particular piece of music at all, unless it happened to be one that you composed yourself and memorized. The goodness of any representation is inextricable from its purpose. Musical notation has proven to be an excellent way to transmit musical ideas from one person to another, across space and time.

So are mathematical symbols and equations an awful representation of nature. — Rich

Again, it depends on one's purpose. For understanding certain relations and making predictions accordingly, mathematical symbols and equations are very good representations of nature. Some of them are even quite beautiful in their simplicity and elegance.

It is interesting that Bergson used music as his analog for duration ... — Rich

Sound in general is a good analog for duration, since there can be no sound at all in a timeless instant. (There can be no light, either, but somehow that is not as intuitive; probably because we have static photographs that remain meaningful.) A collection of sounds only qualifies as music because of the relations among them over time.

Actually, the set of natural numbers is countable by definition, as in mathematics a countable set is defined as a set with the same cardinality as some subset of the set of natural numbers. — Michael

Indeed, but @Metaphysician Undercover only accepts this definition if we divorce it from the colloquial meaning of "countable," which according to him applies only to finite sets small enough that someone can actually finish counting all of their members. He would prefer a different term altogether for the standard set-theoretical concept, like "foozlable" as suggested by @fishfry.

Maybe we should just go with "denumerable" or "enumerable." My dictionary says that these two words both always and only mean "capable of being put into one-to-one correspondence with the positive integers." Now I wish that I had looked them up several thread pages ago ... :(

It is like how you (as far as I can tell) take a deeply theist reading of Peirce, and I the exact opposite. But at the end of the day, the philosophy itself seems strong enough to transcend either grounding. So I can argue even angrily against any hint of theism while also conceding that it is still "reasonable" in a certain light. — apokrisis

Agreed, and likewise. I certainly do not believe that one must share all of a thinker's presuppositions and commitments in order to understand his/her thought and make fruitful use of it. For example, Peirce's theism was far less traditional and institutional than my own, which is probably why you and others can run with him quite a long way without invoking God at all. I have benefited from many of your insights, even if you are completely wrong about this one very crucial detail. :D

I'm used to a robust level of discussion in academic debate. — apokrisis

But this is an online forum, not an academic debate. Oops, putting it that way is likely to have the opposite effect of what I intend ... :s

Look, you and @fishfry are two of my favorite participants here, so I hope that we can all dispense with any unpleasantness and get on with what has already been a fascinating and helpful thread, at least from where I sit.

I've started reading the Zalamea paper and I'm spend some time with it ... It's good stuff. — fishfry

Agreed, and you probably understand a lot more of the non-Peircean content than I do. :)

If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. — fishfry

Perhaps, but it seems to me that we have then already conceded that the real numbers do not and cannot constitute a true continuum. They are now just labels that we have assigned to particular locations along the line, not parts of the line itself.

A point is that which has no part. — fishfry

Right, and a continuum is "that of which every part has parts of the same kind," so obviously a continuous line cannot contain any points. When we mark a point on a line, we introduce a discontinuity.

And a better paper on the Peircean project is probably... — apokrisis

That is indeed a terrific paper, but it gets pretty technical and might be tough to follow for someone not already acquainted with Peirce's thought. Thanks for posting the link, though.

Can you explain (so that a philosophical simpleton like me could understand it) how mathematics has failed to successfully deal with continuity? — fishfry

That is really what my first question is asking. @Rich seems pretty convinced, but I am still trying to make up my own mind, especially since I adhere to Peirce's broad definition of mathematics as the science of drawing necessary conclusions about ideal states of affairs. You mentioned topology specifically, and that is precisely where Peirce turned during the last years of his life.

When you divide a line at a point, the point stays with one segment and not the other. — fishfry

Peirce disagreed with this; he argued that when you divide a line at a point, there are now two points - one goes with each segment. This is because the line does not consist of points and cannot be divided into points; only smaller and smaller lines. Between any two actual points marked on a line, there is an inexhaustible supply of potential points, because it is only when we mark them that they exist at all.

There is no a priori way of determining anything about reality ... — tom

Is there an a posteriori way of determining whether there are any real continua vs. everything (including space and time) being discrete?

Your insinuation that the real numbers cannot somehow model the physical continuum is rather odd. — tom

As I stated in the other thread:

Of course mathematics can and does model a continuum. However, the accuracy and usefulness of such a model depend entirely on its purpose, and that is what guides the modeler's judgments about which parts and relations within the actual situation are significant enough to include. — aletheist

As a structural engineer, I analyze continuous things using discrete models (i.e., finite elements) all the time, and it works just fine for that purpose. Of course, I also apply safety factors to the results, since I am not interested in having the underlying theory falsified.

As I asked in the OP, is it possible to determine whether there are any real continua vs. everything (including space and time) being discrete? If so, how should we go about it?

You haven't stipulated any reasonable definition of countable. — Metaphysician Undercover

Your opinion is duly noted. :-}

But if you want to prove any of these assertions, you need to justify them. — Metaphysician Undercover

Why would I want or need to "prove" or "justify" a generic definition of "x-able" that I have (repeatedly) stipulated?

You appear to be making a category error. "Counting" is an activity of the subject, "countable" is a property of the object. — Metaphysician Undercover

More nonsense. You are the one who wants to define "countable" entirely on the basis of whether it is actually possible for a subject to finish "counting" the object.

If you proceed from what is known about a part, to make a conclusion about the whole, then you commit the fallacy of composition. — Metaphysician Undercover

When I say that elephants are thinkable, or that earth's atmosphere is breathable, or that earth's surface is walkable, or that the natural numbers are countable, I am not reasoning from part to whole. I am not referring to any particular part of each thing, I am stating a general property of each thing. Elephants in general are thinkable, earth's atmosphere in general is breathable, earth's surface in general is walkable, and the natural numbers in general are countable. This is a perfectly legitimate and common use of language.

So, there's no such thing as fallacious reasoning then?! — TheMadFool

Where did I say that? I was addressing your specific examples, which are indeed fallacious as deductive reasonings, but valid as retroductive or inductive reasonings. It all boils down to the purpose of the reasoning. If you want to guarantee that you will only derive true conclusions from true premisses, then you go with deduction. If you merely want to formulate a plausible hypothesis, then you go with retroduction. If you want to test a hypothesis, then you go with induction, but only after deductively explicating it to derive experimental predictions.

If we regard the physical world as the 'cause' of our phenomenal experiences, then the existence of the physical world - thus defined - is an assumption, based abductively (and hence partly subjectively) on our experiences. — andrewk

Yes, exactly. It is a retroductive hypothesis that provides a very plausible explanation for our experiences, which we can then deductively explicate and inductively evaluate by means of our subsequent experiences. However, we cannot know with absolute certainty that our experiences really do correspond to a physical world; just that they correspond to some kind of external world that reacts with us by resisting our actions and our wills, which is what "existence" means.

I agree with Kant that he was not an Idealist (in the sense typically applied to George Berkeley, although Berkeley called himself an Immaterialist). — andrewk

Peirce was intimately familiar with Kant's writings, but eventually chose to call himself an objective idealist instead. He understood "the physical law as derived and special, the psychical law alone as primordial," and stated, "The one intelligible theory of the universe is objective idealism, that matter is effete mind, inveterate habits becoming physical laws." In other words, he held "matter to be mere specialized and partially deadened mind," and that "matter is merely mind deadened by the development of habit," such that "dead matter would be merely the final result of the complete induration of habit reducing the free play of feeling and the brute irrationality of effort to complete death."

He believed in the existence of a physical world. He just also believed that it was unknowable. — andrewk

By contrast, Peirce believed that the physical world is knowable, precisely because matter and mind differ merely in degree, rather than kind. In fact, Peirce insisted that it was absurd to call anything "real" that was unknowable; the real is precisely that which would be known by an infinite community of investigators after an indefinite process of inquiry.

I believe on analysis Pierce's definition is impossible to implement, e.g. defining properties independent of a person or a group of people. — Rich

The issue is not defining properties independent of a person or group of people, it is things having properties independent of what any person or group of people thinks about it.

My dreams are very real to me. — Rich

In my view (and Peirce's), they are either real (full stop), or they are not. Something is real if and only if has properties that do not depend on what any one person or finite group of people think about it. The properties of your dreams depend entirely on what you think about them, so they are not real by this definition.

Given the reals, please count the three members that come after 0.999... and tell me what they are, or even what the next number is so we know we have only two of them. — tom

Why are you asking me? You and I agree that the real numbers are not countable.

To be able to do something, is to be able to complete that task. — Metaphysician Undercover

See, this appears like nonsense to me. One can be able to do something on an ongoing basis, such that whether one is able to complete that task is irrelevant. I am able to be thinking about elephants, so elephants are thinkable. I am able to be breathing earth's atmosphere, so earth's atmosphere is breathable. I am able to to be walking on the earth, so the earth is walkable. And I am able to be counting the natural numbers, so the natural numbers are countable.

If you do not finish counting something then it is not counted. If you cannot finish counting it then it cannot be counted. — Metaphysician Undercover

This right here is where we disagree. To count something is not the same as to finish counting it. Being able to count something is not the same as being able to finish counting something. In other words ...

Everyone who is not being deliberately obtuse understands what countable means - it means you can count elements of the set. No one, unless they are being deliberately obtuse, thinks that this fact has any bearing on whether anyone would be willing to embark on counting all the members of a very large or even infinite set. — tom

We simply have different non-technical definitions of "countable." It is not the case that yours is true and mine is false, or vice-versa; they are just different.

If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it. — fishfry

I appreciate the sensibility that you have tried to bring to this discussion, but I still have to comment on this (again). I agree that neither you, nor I, nor anyone else can actually count that many things. However, this is irrelevant to what we have been discussing, since mathematics is the science of drawing necessary conclusions about ideal states of affairs. An immortal being in an eternal universe could actually count that many things (and more), and this is basically what I mean when I say that they are countable in principle.

I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around. — fishfry

Yeah, I kind of opened the fire hose on you there; sorry about that. There is a handy online dictionary of Peirce quotes on a lot of different topics; going through a few of the most relevant terms might at least give you a taste of his thought on these matters.

It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. — Metaphysician Undercover

Apparently you are in such a big hurry to reply that you are not even bothering to pay attention to what I actually post. In this case, you are mixing up the first two definitions that I so carefully spelled out. The first one, which directly quoted @fishfry, is the one from set theory - not my set theory, but standard set theory - and if it helps, we can substitute "foozlable" as he just suggested (again). The second one - the one that I assume you are still criticizing - has nothing to do with set theory at all, as @fishfry helpfully pointed out a while ago. We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no.

It would be false to say that something which is not capable of being counted is countable. — Metaphysician Undercover

Only if it were true that we must be able to finish counting something in order to call it "countable." Your definition requires this; mine does not.

Therefore we can conclude that the set of natural numbers is not countable. — Metaphysician Undercover

And yet set theory explicitly says otherwise.