If the width of 'now' is zero, time never flows. — Devans99

Agreed.

'now' cannot be infinitesimal width because actual infinity is metaphysically and logically impossible — Devans99

As usual, this wrongly confuses "infinitesimal" with "actual infinity." The proper mathematical definition of "infinitesimal" in this context is having length that is non-zero, but shorter than any assignable value relative to an arbitrary unit interval. The proper phenomenological definition of "moment" in this context is a continuous span of time with duration too short for any sensible change to take place.

So 'now' must have a finite, non-zero length which we can call a moment. — Devans99

What is this alleged finite, non-zero duration of "now"? On what rational principle can we go about determining it, rather than just arbitrarily defining it?

Besides one 'indefinite moment' is defined by the preceding. — Devans99

No, "indefinite" quite literally means "not defined" (by anything). We directly perceive the continuous flow of time within the present moment, and then abstract distinct instants that stand in the relations of preceding (earlier than) and following (later than).

But you said that potential infinity has metaphysical actuality. — fishfry

No, I said exactly the opposite of that.

In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"? — Relativist

Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former. — aletheist

Actual infinity corresponds to metaphysical actuality, while potential infinity and mathematical existence correspond to logical possibility.

Since I have found you to be a normally clear-headed and insightful participant here, I fear that your persistence in dealing with @Metaphysician Undercover lately may be producing some unfortunate side effects. :grin:

I disagree, moments are arranged sequently so they must be representable by the real number line or the naturals. — Devans99

No, that only applies to distinct instants, not indefinite moments. In any case, what is the first real or rational number after zero? It straightforwardly begs the question to insist that only the natural numbers can be used here, because by definition they have a first member.

But how is that metaphysical actuality? — fishfry

It is not; as I said, mathematical existence--including the potential infinity of the natural numbers--is not metaphysical actuality, it is logical possibility. From your other comments, I think that we agree on this; perhaps you misread my previous post.

In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"? — Relativist

Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.

Do abstractions exist at all? — Relativist

As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms.

It's time consuming to really get to know him [Peirce]. — Mapping the Medium

This is most certainly true.

Immanuel, or the Son of God, isn't necessarily God. — Ciceronianus the White

Again, not germane to the thread topic, but "Immanuel" means "God with us," not "Son of God." Paul says that Jesus "was in the form of God ... but emptied himself," and calls him "the image of the invisible God" such that "by him all things were created" and "in him all things hold together." The earliest Christian creed was "Jesus is Lord," identifying him with the name typically used for God in place of the unspoken YHWH.

Jesus wasn't referred to as God in the earlier Gospels, or in Paul ... — Ciceronianus the White

Not really germane to the thread topic, but this is clearly false; e.g., see Matthew 1:23, Mark 1:1, Luke 3:21-22, Philippians 2:5-11, and Colossians 1:15-20, just for starters.

Clearly there was a reason Peirce attended the Episcopal Church, a midway between Catholic and Protestant. — Mapping the Medium

Yes, probably because his first wife was Episcopalian--her father was a bishop--and she presumably only agreed to marry him if he converted from the Unitarianism of his family.

I was baptised Lutheran, and recognize the nominalism within it. — Mapping the Medium

I became Lutheran as an adult, and have not encountered anything in its theology that absolutely requires nominalism or rejects scholastic realism. In general, Lutherans are wary of imposing any particular philosophical system.

This is one good reference that may shed some light on his views. — Mapping the Medium

Naturally, I prefer this one.

I searched "conscience charles peirce" on google, and nothing came up — Gregory

Try downloading the combined PDF of the eight-volume Collected Papers and searching for "conscience" there.

... to understand Peirce means to understand that reality... 'Mind'.. is immanent AND transcendent. Whether you want to call that mind 'God' is left up to the interpreter. — Mapping the Medium

According to Peirce, there is indeed both immanent mind and transcendent mind, but only the latter is properly called God.

I do not mean by God a being merely "immanent in Nature," but I mean that Being who has created every content of the world of ideal possibilities, of the world of physical facts, and the world of all minds, without any exception whatever. — Peirce, R 843, 1908

In case there is any doubt about what he was denying here, it turns out that he wrote the definition of "immanent" for the Century Dictionary, which includes the following.

In modem philosophy the word is applied to the operations of a creator conceived as in organic connection with the creation, and to such a creator himself, as opposed to a transient or transcendent creating and creator from whom the creation is conceived as separated. The doctrine of an immanent deity does not necessarily imply that the world, or the soul of the world, is God, but only that it either is or is in God. — Peirce, Century Dictionary

If God is not immanent, then by this definition He is necessarily transcendent; both pantheism (the world is God) and panentheism (the world is in God) are ruled out. It is therefore untenable to ascribe either of these views to Peirce, as some scholars wrongly do; he was a Protestant Christian theist, although admittedly not a traditionally orthodox one.

Peirce actually sought to understand the trinity as 'Logos'. — Mapping the Medium

Citation, please. The first verses of the Gospel of John explicitly identify the Logos (Word) with only one Person of the Trinity, the Son who became flesh and dwelt among us. The best treatment of the Trinity from a Peircean standpoint that I have come across so far is Andrew Robinson's 2010 book, God and the World of Signs: Trinity, Evolution, and the Metaphysical Semiotics of C. S. Peirce.

Regarding Christian theology, I was referring to these incorrect statements.

Christians try to get out of moral responsibility by saying God "became sin" and destroyed it. All you have to do is accept that he did and your responsibility is removed. — Gregory

As for Peirce, do your own homework.

Ah, someone else who is completely unfamiliar with Peirce's writings, not to mention Christian theology. Unwilling to understand, yet eager to reject.

Easy to say, impossible to show (apparently). You seem locked into specific conceptual dichotomies of mind/matter, thought/reality, and subjective/objective, such that you are unwilling/unable to consider a different point of view.

The world clearly exists as a reality, not as thought. — Gregory

False dichotomy. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it. This does not entail that the real is independent of thought in general.

And why has he been called an objective idealist instead of a subjective one if he thinks the world is subjective? — Gregory

He did not think that the world is subjective; see the definition of "real" above. Besides being an objective idealist, he was also an extreme scholastic realist.

Indefinite, means unlimited, which is the same as infinite. — Metaphysician Undercover

No, that is not what "indefinite" means in this context.

"Infinite" implies unlimited, while "infinitesimal" implies a limit. — Metaphysician Undercover

No, that is not what "infinitesimal" means in this context.

"Self-deluded" might be accurate, but "deceptive" might actually be more precise, as described below. — Metaphysician Undercover

No, calling Peirce a materialist (dialectic or otherwise) demonstrates complete unfamiliarity with his actual writings, as evidenced by the persistent refusal to offer any supporting quotes or citations whatsoever.

Are the the Peirce people on this thread saying that we all have a common mind and that mind creates the material world? — Gregory

No, Peirce's view was that mind is primordial, such that "matter is a peculiar sort of mind."

Saying that the boundaries are vague and not distinct, does not say that there are no boundaries. — Metaphysician Undercover

Now this is an example of splitting hairs, so I will rephrase. Infinitesimals are necessarily indefinite, while boundaries are necessarily distinct, so infinitesimals have no boundaries.

Someone like Peirce, can say "I am not materialist, my metaphysics is objective idealism", and still offer us a metaphysics based in materialist principles. — Metaphysician Undercover

Ah, so now you are claiming that Peirce was either self-deluded or a liar. Time to show your work--provide quotes demonstrating that his metaphysics was based on materialist principles, or just admit that you are not familiar with his thought and are just making stuff up.

Are you saying Peirce is a subjective idealism instead of an objective idealist? — Gregory

Absolutely not. Mind is not confined to human minds.

It appears that your concept of mind is too narrow. "The world is mind" does not entail that "we create the world."

Objective idealism says we create the world and it's real. George Berkeley believed in the former, while Hegel, Royce, Peirce, and Giovanni Gentilies probably can be said to be objective idealists — Gregory

That is painting with far too broad a brush. Peirce's objective idealism does not say that we create the world, it describes "the physical law as derived and special, the psychical law as primordial" such that "matter is effete mind, inveterate habits becoming physical laws" (CP 6.24-25; 1891).

Infinitesimals are within a continuous medium, and they also require boundaries. — Metaphysician Undercover

Are you just not paying attention? Infinitesimals do not have distinct boundaries, which is why the principle of excluded middle does not apply to them.

Exactly as I said, Peirce allows for violation of the law of non-contradiction. — Metaphysician Undercover

Are you just not paying attention? He held that the principle of contradiction (not the same as LNC) does not apply to that which is vague/indefinite.

Therefore he is dialetheist, and in my judgement, dialectical materialist. — Metaphysician Undercover

Are you just not paying attention? Your judgment is incorrect; Peirce vehemently rejected materialism, explicitly identifying his metaphysics as objective idealism.

Brilliant insight! You have amply demonstrated in other threads that you are not interested in counter arguments. Your mind is already firmly made up that reality is entirely discrete and finite, and any approach to mathematics that is not consistent with this dogmatic assumption is to be discarded.

The thread title should be "Devans99's Mistake." Your constant grinding of this particular ax became tiresome long ago. Saying the same thing over and over, and expecting a different result, is one definition of insanity.

No one gets to choose a "version of reality," because by definition reality is as it is regardless of what anyone thinks about it. Adopting finite discrete arithmetic as the sole or primary basis for your entire metaphysics strikes me as extremely naive. I hope that someday you will be more open to alternatives.

The proof of irrational numbers is common sense. We have empirical evidence for atoms and the curvature of spacetime. So these things are in agreement with common sense. — Devans99

We clearly have very different definitions of "common sense."

But as seekers of a truthful explanation of our reality, we have to make choices between incompatible branches of mathematics. I'm unwilling to discard arithmetic from my set of choices of valid mathematics. — Devans99

Who said anything about discarding arithmetic? It is very useful for very many purposes, especially those encountered in everyday life, which generally involve dealing with finite quantities of discrete things. A different approach is required to handle potentially infinite sets, and yet another is required for true continuity. Whether this accords with "common sense" or not, it is the reality.

1) Do you think that consciousness is 'real'? — Mapping the Medium

Yes, although before going any farther we need to establish what we mean in this context by "consciousness" and "real."

2) If so, do you think it can be explained/described numerically? — Mapping the Medium

I am not sure what that would even mean, but like anything else, I think that it can be explained/described mathematically--i.e., with a retroductive hypothesis that we can then deductively explicate and inductively evaluate. The subject matter of mathematics is much broader than just numbers.

4) Considering the above questions, which came first, consciousness, numbers, or semiotics? — Mapping the Medium

Again, we need to define "consciousness," and numbers are strictly hypothetical. Semiotics is a relatively recent science, so it clearly came last; but if you meant to say semeiosis, the real process that semiotics studies, then I am inclined to believe that it came first.

5) How do all of the above questions factor into time, evolution, and ever-changing linguistics? — Mapping the Medium

My current working hypothesis is that time is a manifestation of semeiosis, the ongoing evolution of the universe as dynamical objects determine sign tokens to determine dynamical interpretants. Linguistics is a special science that studies actual languages, while semiotics is a normative science that studies the nature of signs in general (speculative grammar), good vs. bad reasoning (logical critic), and methods for obtaining true beliefs (speculative rhetoric).

If parts of maths adopt axioms that depart from common sense, then I have to disregard those parts when searching for a description of our reality. — Devans99

Common sense tells us that common sense is highly fallible. Some developments in mathematics and science over the centuries are highly counterintuitive, and if we had insisted on sticking with common sense, we would still be misunderstanding reality. Is it common sense that there are numbers incapable of being calculated as fractions of integers? Or that matter consists of atoms that in turn consist of varying quantities of protons, neutrons, and electrons? Or that gravity is the curvature of spacetime, rather than a direct force of attraction between massive bodies?

It is only possible to hold a belief in one of these two incompatible parts of math. — Devans99

Nonsense, there is no single set of mathematical assumptions that perfectly matches reality--just different models that are useful for different purposes.

I know what you mean about clones! Hopefully you can at least digest the two short articles about SDG and SIA. Browsing Moore's single volume of Peirce's writings on philosophy of mathematics might be enough to give you the gist of his overall approach and help you decide whether delving deeper is worth the trouble.

So the point is useless and that's why I said Peirce dismisses it. — Metaphysician Undercover

Citation, please. On the contrary, Peirce does not dismiss points/instants, he clarifies that they are creations of thought rather than real constituents of lines/time.

The existence of infinitesimals in the medium requires that there are natural boundaries. — Metaphysician Undercover

No, the only boundaries within a continuous medium are the artificial ones that we arbitrarily insert at finite intervals for some particular purpose, such as measurement.

Peirce is a dialectical materialist, or dialetheist, one who allows for the law of non-contradiction to be violated. — Metaphysician Undercover

Citation, please. On the contrary, according to his own words Peirce is an objective idealist for whom the principle of non-contradiction does not apply to that which is vague/indefinite and the principle of excluded middle does not apply to that which is general/continuous. In accordance with the latter, he is now recognized as the first person ever to develop truth tables for a rudimentary three-valued logic--true, false, and the limit between truth and falsity.

The problem with the articles you linked is that in both cases, a wrong assumption is made at the start of discourse: — Devans99

There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things.

I happen to strongly believe that the LEM holds for our universe and indeed all possible universes. — Devans99

LEM holds for anything that is determinate, including anything that is discrete; but it does not hold for anything that is indeterminate, including anything that is truly continuous. There is nothing illegitimate about intuitionistic logic.

I also strongly believe there is no nonzero x such that x^2=0. — Devans99

What you believe is irrelevant. There is nothing self-contradictory about defining an infinitesimal as that which is not itself equal to zero, but whose squares and higher powers are equal to zero; and it turns out to be quite useful.

Why should I invest time and effort learning subjects that are based on wrong assumptions? All 'knowledge' I'd acquire in doing so would be inherently unsound. — Devans99

The problem is treating assumptions other than your own as indubitably wrong and refusing even to entertain them, which is a textbook example of sheer dogmatism. Unless you consider yourself to be infallible, you might want to try opening your mind a bit.

The teaching of math in the US is stupid by order of the government. — fishfry

I laughed out loud at this. It reminds me of my favorite line from the movie, The Incredibles 2, when Bob is trying to help Dash with his homework: "Why would they change math? Math is math!"

So just as I call modern constructivism Brouwers revenge, I can call SIA Peirce's revenge. — fishfry

Heh, I like it! I am an engineer, not a mathematician, so I would welcome your thoughts on SDG/SIA--although they probably belong in a new thread.

Do you happen to know how SIA relates to constructive math? They both deny excluded middle as I understand it. — fishfry

Not really, but I suspect that they have different reasons for denying excluded middle. You might be interested in John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. The first half covers the history, while the second half consists of chapters specifically on topology, category/topos theory, NSA, constructivism/intuitionism, and SDG/SIA. A new version just came out with an even longer title, but best I can tell the only significant change is the addition of several appendices on various topics.

What's R 300? Where can I find Peirce talking about calculus? — fishfry

R 300 means manuscript number 300 as cataloged by Richard S. Robin in the 1960s. That particular text is incomplete and largely unpublished, but there is a transcription online. Peirce does not actually talk about calculus much in it, and I honestly do not know where else in his voluminous writings he might have done so in any detail. You might find some leads in the Robin catalog, and then you can browse through images of the actual manuscripts.

I don't suppose I could ask for a simple explanation of what this phrase means? Is it anything like the quotient of dy and dx being the derivative when dy and dx "become" zero but aren't actually zero, as Newton thought of it? — fishfry

Here is what Peirce wrote right before the sentence that I quoted.

Accepting the common-sense notion [of time], then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. — Peirce, R 300, 1908

Presumably the "false continuity" that he had in mind was that of Cantor. Does this help at all?

Why are there so many Peirceans on this forum? I never hear about him anywhere else but his ideas are incredibly interesting. — fishfry

I am not sure that there are really so many of us here, but I obviously agree with that last part. The problem is that Peirce never wrote a magnum opus spelling out his entire philosophical system, or for that matter any significant portion of it.

The Collected Papers (eight volumes) are arranged topically, rather than chronologically, with some rather misleading results because his thought definitely evolved over time. Unfortunately, the comprehensive critical edition being prepared by the Peirce Edition Project has been stuck for ten years after producing only eight volumes of thirtyish planned, primarily due to chronic lack of funding. The Essential Peirce (two volumes) is probably the most accessible compilation, but emphasizes general philosophy. Two specifically mathematical compilations--Carolyn Eisele's New Elements of Mathematics (four volumes in five) and Matthew E. Moore's Philosophy of Mathematics: Selected Writings--are probably more relevant to you.

Do you agree with this, there is no infinitesimal without a boundary to separate it from others? — Metaphysician Undercover

No, I do not agree with this, and neither did Peirce. Infinitesimals/moments are indefinite, not distinct. The principle of excluded middle does not apply to them. Points/instants are arbitrarily inserted as boundaries between continuous segments/lapses of finite length.

Where do you get the idea that Peirce thought there were infinitesimals which are not created by those arbitrary divisions? — Metaphysician Undercover

Again, his view was that infinitesimals/moments are real, while points/instants are artificial creations. I have already provided several quotes from his writings to support this summary. Unless you can produce citations to the contrary, we have nothing more to discuss here.

I have come across some arguments that teaching calculus using SDG/SIA--grounded in category theory, rather than set theory--is more effective than either standard or non-standard approaches. In any case, scholars of Peirce's mathematical thought seem to agree that it comes much closer to being a rigorous implementation of his conceptions of infinitesimals and continuity than NSA. For one thing, the principle of excluded middle does not apply--i.e., the logic of SDG/SIA is intuitionistic, rather than classical--which is exactly what he consistently maintained about anything that is general, including anything that is truly continuous. For another, functions in SDG/SIA "are differentiable arbitrarily many times" (Bell), consistent with this statement.

Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. — Peirce, R 300, 1908

Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. As demonstrated in the op, ∞ leads to logical absurdities, so logically 1/∞ must be absurd too. — Devans99

There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/∞. Again, if you truly want to understand, please read one or both of the short articles that I linked. If you prefer to remain ignorant, carry on.

Again, Peirce himself considered his realism to be "extreme." That does not seem very "nuanced" to me. :smile:

They are distinct, like a movie plays at 60 frames a second, each frame a time slice. There is nothing continuous about that. — Devans99

That is exactly what I described as discrete time, not what you described. Each individual frame of such a movie corresponds to an instantaneous state, with zero duration, arranged at a fixed and finite interval of 1/60th of a second. There is no flow of time between the frames, just a leap from one to the next.

So infinitesimals cannot be the constituents of time because all time intervals would have zero length and time would not flow from one moment to the next. — Devans99

Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. The present moment includes an infinitesimal portion of the past and an infinitesimal portion of the future, which is why time does flow from one moment to the next.

If you truly want to understand the mathematics of infinitesimals, I recommend learning about synthetic differential geometry, also called smooth infinitesimal analysis. These links are to excellent brief introductions by Sergio Fabi and John Bell, respectively.

Discrete time would consist of discrete, non-zero, non-infinitesimal time slices - so they would have a duration. — Devans99

In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer.

On the other hand, if time was composed of infinitesimal time slices, then each fixed period of time would be composed of 1/∞=0 length time segments giving a zero total length for all elapsed intervals. Which cannot be right. — Devans99

The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals.