There are components of the future that are determinate, for example: the positions of the planets with respect to the sun, at this exact time tomorrow. Does this determinacy mean it exists? — Relativist

Peirce's view was that nothing in the future is strictly determinate; i.e., he rejected determinism, which he usually called necessitarianism, instead embracing the reality of absolute chance. Again, it is better to say that if the relevant circumstances were to remain unchanged, then the planets would appear in certain positions with respect to the sun at a specified time tomorrow.

He's embracing past tense as reality, yet denies future tense as reality, right? — 3017amen

No, this conflates reality with existence. The past exists, because it is determinate; the future does not (yet) exist, because it is indeterminate. However, there are real possibilities and real (conditional) necessities in the future, just no real actualities. The present is when some of those future possibilities and necessities become past actualities. The "arrow of time" reflects how the universe is proceeding from an ideal state of complete indeterminacy at the hypothetical commencement of all time, when everything would have been in the future, toward an ideal state of complete determinacy at the hypothetical completion of all time, when everything would be in the past.

If someone on the west coast is making a call to someone on the east coast, are they not talking to someone in the future? — 3017amen

No, both of them are talking to each other in the present. The difference in their spatial locations has no bearing on their temporal relation. The fact that an east coast clock reads three hours later than a west coast clock is an arbitrary convention of how we mark and measure time, and reflects nothing about the real nature of time itself.

If finite figures have an infinity of points, that is paradoxical. — Gregory

Finite figures have no actual points, but infinitely many potential points. Dimensionless points are not parts of the figures, they are something that we artificially impose on them for particular purposes, such as marking and measuring. Every part of a one-dimensional line is a one-dimensional line, every part of a two-dimensional surface is a two-dimensional surface, and every part of a three-dimensional solid is a three-dimensional solid. Figures of lesser dimensionality--a point on a line, a point or a line on a surface, and a point or a line or a surface on a solid--are limits that we create by arbitrarily dividing the whole into parts.

Okay, but with respect to infinity, how does that square with the common paradox of past present and future? — 3017amen

Again, I am not sure what exactly you have in mind here. How would you succinctly summarize "the common paradox of past present and future"?

I thought you said that he denied future tense. — 3017amen

I said that the future does not exist because it is indeterminate; i.e., nothing in the future is actual, it is either possible or (conditionally) necessary, which are two sides of the same modal coin. That is why I rephrased your declarative proposition in future tense as a subjunctive conditional. Among the popular modern theories of time, Peirce's view seems closest to the "growing block universe."

Or maybe he's thinking that everything is a subjective illusion. — 3017amen

Definitely not; again, there are realities which are as they are regardless of what any individual mind or finite group of minds thinks about it.

However, I'm not totally on-board with his esoteric definition of metaphysics. — 3017amen

I am not sure what you mean by "his esoteric definition of metaphysics." For Peirce, "Metaphysics consists in the results of the absolute acceptance of logical principles not merely as regulatively valid, but as truths of being." Accordingly, "Just as the logical verb with its signification reappears in metaphysics as a Quality, an ens having a Nature as its mode of being, and as a logical individual subject reappears in metaphysics as a Thing, an ens having Existence as its mode of being, so the logical reason, or premise, reappears in metaphysics as a Reason, an ens having a Reality, consisting in a ruling both of the outward and of the inward world, as its mode of being. The being of the quality lies wholly in itself, the being of the thing lies in opposition to other things, the being of the reason lies in its bringing qualities and things together." These three modes of being correspond respectively to abstract characters as denoted by general terms, concrete things as denoted by quantified variables, and prescissive facts as signified by propositions.

What is your take on his Objective Idealism? — 3017amen

I find it persuasive. "The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws," such that "what we call matter is not completely dead, but is merely mind hidebound with habits." Physical motion is thus a degenerate form of psychical thought or semeiosis. "Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible."

Is he saying that it is not reasonable, through induction, that the sun will rise tomorrow? — 3017amen

No, it is a reasonable but fallible prediction based on our knowledge of the real laws that govern existents. Propositions about the future are more appropriately stated as subjunctive conditionals--if the relevant circumstances were to remain unchanged, then the sun would rise tomorrow.

The point is that what exists for us is whatever can react with us. If it is utterly impossible for B to react with us, then we have no basis for claiming that B exists.

That seems problematic, since it makes existence relative, and both A and B have being. — Relativist

No, it simply maintains the definition of existence as reaction with other things in the environment. Existence is a special kind of reality, which is a special kind of being. Everything that exists is real, but there are realities that do not exist (e.g., some abstractions); and everything that is real has being, but there are beings that are not real (e.g., fictions).

From A's perspective, does B exist? — Relativist

I am inclined to say no, since there is nothing in B that reacts with anything in A.

Is B real? — Relativist

I am inclined to say yes, since B is as it is regardless of what anyone thinks about it.

Clocks are just the way we take the abstract and make it concrete. — 3017amen

The clock itself is concrete, but the time that it marks and measures is still abstract, and the units by which it marks and measures time are arbitrarily defined.

From ancient sun dials to early Americans harvest season, to modern day atomic clocks, time has in effect, been shortened in it's perception of change. — 3017amen

So the duration of time itself has not decreased throughout history, only the smallest measurable unit of time; i.e., we can mark and measure time more precisely than our ancestors.

But yeah in an anecdotal way I've often wondered about that psychological phenomenon too, as to why time seems to go by faster as we age. — 3017amen

My longstanding hypothesis about this is that as each day passes, it becomes a smaller and smaller fraction of one's entire lifespan. One year out of ten is a sizable chunk, but one year out of fifty--not so much.

The assumption that every aspect of the universe can be so pixeled assumes his conclusion. — jgill

In other words, there is no actual infinity of discrete objects; but this does not rule out real continuity in the universe, such as that of time and space, which are not composed of distinct parts.

I am only vaguely familiar with Plantinga's approach, but the discussion and examples so far remind me of the non-foundationalist pragmatism of Charles Sanders Peirce. He sometimes preferred to call it "pragmaticism" in order to distinguish his realist version from the more popular nominalism of his good friend William James.

Our direct perception of things and events prompts us to make involuntary perceptual judgments that are quite fallible. We cannot help believing them at first, and they lead us to anticipate certain other things and events going forward. When those predictions are fulfilled in our subsequent experience, the belief--which is really a habit of conduct, even when expressed as a proposition--is corroborated; when they are confounded, it is falsified. In other words, this is just our everyday implementation of the scientific method:

1. Retroduction, formulating a plausible hypothesis.
2. Deduction, working out its necessary consequences.
3. Induction, evaluating whether those outcomes are realized.

Moreover, Peirce held that we have a direct perception of God. He somewhat famously wrote an article near the end of his life that presented "A Neglected Argument for the Reality of God" as a process of retroduction, rather than deduction as employed in most classical "proofs."

Can you explain time in better terms? Maybe in less abstract terms, your choice. — 3017amen

I am attempting to spell out my Peirce-inspired ideas about time in another thread, so I suggest that we continue this conversation over there.

What we perceive we perceive as a present event-as going on now. — 3017amen

Peirce makes another helpful distinction between an event as a definite and entire change vs. a state of change as "indefinitely gradual"; i.e., strictly continuous. What they have in common is that both are realized only at a general determination of time (lapse), rather than at an individual determination of time (instant). For any event, the states of things at the commencement and completion of the corresponding lapse are "incompossible facts"; but during the lapse itself, neither of them is realized. The present is always just such a state of change, because various events are constantly in progress throughout the universe.

However, when we perceive such an event, that event is no longer in the present. The actual 'present' exists only for an instant; the event becomes memory. It follows that we perceive the past to experience the present. — 3017amen

I suggest instead that we directly perceive the event as it happens in the present, which again is an indefinite lapse (not a distinct instant) during which a state of change is realized. However, we then involuntarily make a perceptual judgment about the event, and this is indeed in retrospect; so all our knowledge is about the past. Peirce even defines the past as "that part of time with which memory is concerned" and the future as "that part of time with which the will is concerned," such that "Events past are recalled by memory supposing they acted on our sense; events to come are anticipated supposing they are subject to our will."

In fact, it could be argued that the past and future truly exist, while the present is only a variable instant. — 3017amen

On the contrary, using Peirce's definition, the past truly exists--it acts on us, and we react to it--but the future does not. The past is determinate, while the future is indeterminate; so the present "is plainly that Nascent State between the Determinate and the Indeterminate." Likewise, the past is actuality, while the future is possibility and (conditional) necessity; so the present "is the Nascent State of the Actual."

Time as concrete is simpler; lunar cycles, harvest season, the sundial, the clock, stop watch, etc. — 3017amen

These are not real parts of time itself, but rather arbitrary intervals between states of things that are similar and regular enough for us to use them conveniently to mark and measure the passage of time.

It's interesting to note that the duration of time has seemingly decreased throughout history. — 3017amen

According to whom? What exactly does "duration" mean in this context?

Eternity equals no resolution. — 3017amen

Eternity usually means timelessness, or perhaps infinite time. I still see no connection with incompleteness and incomputability.

I explained earlier what the definition of abstract time is viz the concept of eternity. — 3017amen

Not in this thread, as far as I can tell.

Is that then, abstract and/or eternal? — 3017amen

What do you mean by "abstract" in this context? The present is certainly not eternal.

Gödel and Turing taught us about mathematics being incomplete & incomputable, and never ending, similar to irrational numbers and the modal logic paradox. — 3017amen

I still do not see what those mathematical results have to do with "the concept of time and eternity."

With respect to time, another paradox of trying to define time also appears incomplete in its 'eternal way' of trying to measure same. — 3017amen

Sorry, I do not understand this sentence.

You may have seen this before: — 3017amen

Thanks for the video. To me, the paradoxes identified by Aristotle and McTaggart are resolved by recognizing that the principle of excluded middle only applies to individuals, while time is continuous and therefore general. There are no discrete instants within time itself, they are artificial creations that we impose for the purpose of marking and measuring time. The present moment is always indefinite, blending seamlessly into the immediately past and future moments, yet itself neither past nor future.

The question remains: do immaterial objects exist? If so, what does it mean to exist? — Relativist

From a metaphysical standpoint, Charles Sanders Peirce drew a helpful distinction between reality and existence. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while existence is reaction with other like things in the environment. Accordingly, abstractions and other immaterial objects do not exist, but some of them are nevertheless real.

Does Spider-Man exist? Do all fictions, past present, and future exist? What about possible fictions that never get authored? — Relativist

Metaphysically, Spider-Man and other fictions do not exist and are not real. However, they "exist" in logic where we redefine the scope of that term to a certain universe of discourse, rather than reality itself. Within the fictional world created by certain comic books, movies, etc. there exists an x such that x is Spider-Man; within the real world, no such x exists. On the other hand, those concrete comic books, movies, etc. obviously do exist metaphysically.

Think of time itself, as being both abstract and concrete. (Does time exist? And how does it exist, abstractly?) — 3017amen

What does it mean to say that time is somehow "both abstract and concrete"? I suggest that time is another example of something that does not exist, but is real. It is a law that governs existents, rather than an existent that reacts with other existents.

Time exists as a relation between states of affairs. I don't believe abstraction exist independently of states of affairs. — Relativist

I prefer Peirce's definition: "Time is a certain general respect relative to different determinations of which states of things otherwise impossible may be realized. Namely, if P and Q are two logically possible states of things, (abstraction being made of time) but are logically incompossible, they may be realized in respect to different determinations of time." Or as he put it elsewhere: "Time is that diversity of existence whereby that which is existentially a subject is enabled to receive contrary determinations in existence," such that "a real event" is "an existential junction of incompossible facts."

What does state of affairs mean? — 3017amen

According to Peirce, "A state of things is an abstract constituent part of reality, of such a nature that a proposition is needed to represent it." Moreover, "A fact is so highly a prescissively abstract state of things, that it can be wholly represented in a simple proposition." As I suggested recently in another thread, every proposition signifies a state of things by attributing abstract characters to concrete things, and every true proposition signifies a fact. Again, the universe of discourse matters--e.g., it is a fact that Spider-Man wears a mask and shoots webs within the fictional Marvel world.

Have you considered the concept of time and eternity from Gödel (self-reference), and Turing? Or even the paradox of time itself? — 3017amen

Could you please summarize what you have in mind as "the concept of time and eternity from Gödel (self-reference), and Turing," as well as "the paradox of time itself"? Thanks in advance.

I seem to have a problem with a proposition being characterized as true or false because, by my understanding, a proposition is not definitive. Its values of truth and falsity are potential. To me, if the potentiality is verified, then it becomes an axiom ... Is my explanation sound? — BrianW

No, that is not how "proposition" and "axiom" are typically defined in logic and philosophy. There are all kinds of true propositions, only a few of which are considered to be axioms. "My PF screen name is aletheist" is a true proposition, but it is certainly not an axiom.

I would not object to that usage of the word "fact". Though I think that it can be used either way unproblematically. — Douglas Alan

Understood, and people do routinely use "fact" as a synonym for "true proposition." I just find it helpful to maintain a careful distinction between a true proposition and the state of things that it represents for an interpreter thereof by reserving "fact" for the latter.

A fact is a true proposition. — Douglas Alan

I propose instead: A fact is the state of things that is signified by a true proposition.

Can a proposition be true? — BrianW

A proposition is traditionally defined as a sign that can be true or false, in contrast to a term or an argument.

Furthermore, I actually agree with a lot that Peirce has said. — Metaphysician Undercover

How would you know? You have not demonstrated any familiarity with his voluminous writings.

Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions ... — Metaphysician Undercover

Nonsense. As usual, bare assertion with no basis in fact.

... because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this. — Metaphysician Undercover

I refuse to acknowledge this because it is blatantly false.

One assertion after the other, without explanation or justification, together with an ignorance of logic; you just reminded me of how difficult it is to hold a discussion with you. — Metaphysician Undercover

LOL! Right back at you.

What we really end up with "typically", is an extremely expensive building designed to protect the engineer from lawsuits. — Metaphysician Undercover

Thanks for confirming that you have no idea how building design and construction actually work.

The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur. — Metaphysician Undercover

History demonstrates otherwise, as @fishfry has pointed out.

If the axioms involved are false, then what follows from them cannot be sound conclusions. — Metaphysician Undercover

The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. If a certain theorem follows necessarily from a particular system of axioms, then it is true within that system.

It should be obvious to you, a structure is built on foundations, not vise versa. — Metaphysician Undercover

I happen to be a practicing structural engineer. What might not be obvious to you and others is that a structure is always designed first, followed by its foundations. We have to establish what needs to be supported before we can go about determining how it will be supported. Often there are multiple options--footings on native soil, footings on special fill materials, footings on improved ground, driven steel piles, driven precast concrete piles, auger-cast grout piles, drilled shafts filled with cast-in-place concrete, etc. Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different?

The philosophy of math might study the foundations, but it does not produce the foundations. — Metaphysician Undercover

A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge.

But you misunderstand philosophy, which seeks to distinguish truth from mere usefulness. — Metaphysician Undercover

Some philosophers define truth as mere usefulness. I am not one of them, and neither was Peirce, although many of them call themselves "pragmatists"--one reason why he eventually started calling himself a pragmaticist instead.

"Usefulness" is judged in relation to a goal or end, and that goal or end may be contrary to truth ... The philosopher seeks to know, without regard for the usefulness of that knowledge, knowing for the sake of knowing. — Metaphysician Undercover

In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it.

It was a bipartisan hearing room. — 3017amen

The Republicans in the room were not allowed to call any witnesses of their own, and were restricted in their questioning of the witnesses who did appear.

The Republican's fear retribution from Dumpertrumper because they want to get re=elected. Correct? — 3017amen

Maybe in some cases, definitely not in others; but once again, this is an attempt to discern motives rather than sticking to facts.

The GOP is very emotional indeed! — 3017amen

Says the person who refers to the duly elected President of the United States as "Dumpertrumper."

Explain what is incorrect in stating that this is the first impeachment trial without witnesses? — 3017amen

Nothing, but that is not what you said.

This is the first Impeachment without witnesses. — 3017amen

There were witnesses for the House impeachment--although only certain ones that the Democrats wanted, and many of them testified only in secret--just none for the Senate trial.

If I get a ticket in traffic court and I want to exonerate myself I would bring witnesses. — 3017amen

Or you could come by yourself, and if the police officer who issued the ticket did not show up, then the judge would find you not guilty. In this case, it was not the defendant who primarily wanted to call witnesses, but the prosecutors--because they failed to do a sufficiently thorough job with the grand jury (House) that produced the indictment (impeachment).

Two inferences can be made:
1. They feared witnesses would incriminate and coobberate Dumpertrumper's behavior.
2. They would want to exculpate and thus exonerate their parties leader. — 3017amen

3. They did not believe that additional witnesses would have revealed any new information that would have changed their assessment--President Trump's conduct did not warrant removal from office. Also, your #1 again suggests that emotion--not logic--is guiding your responses.

Nixon and Clinton had a regular trial, right? — 3017amen

Nixon resigned before being impeached, let alone tried; I assume that you meant Andrew Johnson. And no, there is nothing "regular" about any Senate impeachment trial of a sitting president--especially one initiated by the House on a strictly partisan basis.

Even though it wasn't a criminal trial, we all wanted witnesses in order to help determine incriminating or exculpatory evidence. The mere fact there was obstruction of documents and witnesses, suggests a Modus Tollens type of inference. — 3017amen

Impeachment is supposed to be a rare and serious matter. Gathering evidence to support taking such action is the responsibility of the House of Representatives. If its investigators believed that there was genuine "obstruction of documents and witnesses," then they could (and should) have taken that argument to the courts, since that is where such disputes between the legislative and executive branches are routinely resolved. Besides, two Democrats in the House and Mitt Romney in the Senate acknowledged that President Trump was not guilty of obstructing Congress.

This is the first Impeachment without witnesses. — 3017amen

No, it is the first Senate trial for removal from office without witnesses; again, please do not conflate the two distinct steps. It is also the first impeachment without bipartisan support in the House of Representatives, and even Nancy Pelosi once argued against proceeding under such circumstances.

So, what are we left with, reasonable inference you think? — 3017amen

A majority of Senators did not believe that additional witnesses would have revealed any new information that would have changed their assessment--President Trump's conduct did not warrant removal from office. What other inference would be reasonable?

I know that is your interpretation, however, High Crimes and Misdemeanors is simply that, a constitutional interpretation. — 3017amen

Not really, it is a direct quote from the text itself. The Senate has the exclusive power to determine what qualifies.

And would you condone such behavior from any President (requesting foreign assistance for personal political gain-which is in violation of campaign statues.) — 3017amen

Of course not; but in my opinion, the House managers did not establish beyond a reasonable doubt that President Trump requested foreign assistance strictly for personal political gain. That is an attribution of motive, which is a very tricky thing to prove, especially on the basis of a single phone conversation. The issue there is whether an investigation of Burisma and the Bidens was (and still is) properly predicated, in which case the fact that it might also have a personal political benefit for Trump is irrelevant.

Too many people are still conflating impeachment with conviction/removal. Arguably a simple majority of the House of Representatives can impeach any federal official for any reason whatsoever, but as I noted at the very beginning of this thread, a two-thirds majority of the Senate can only constitutionally remove someone from office for "Treason, Bribery, or other high Crimes and Misdemeanors." In this case, a majority of Senators voted that the President was not guilty of anything of that magnitude.

Being immoral, dishonest, and obnoxious--which I have acknowledged to be an accurate description of Donald Trump, which is why I did not and will not vote for him myself--is not sufficient to take the drastic step of overturning the result of a valid election. The American people will have their say in nine months, and so far it looks like the first-ever strictly partisan impeachment of a sitting president has had the opposite of its intended effect on his approval ratings.

Is that an example of Peircean pragmatism or is that an expression of something else? — fishfry

It is consistent with Peircean pragmatism in the sense that one's purpose dictates how one formulates the hypothetical state of things to be explicated--in this case, the particular set of axioms to adopt.

... can you help me to understand Metaphysician Undercover's point of view? — fishfry

I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else. "Existence" has only one meaning, and that is its ontological definition; so asserting mathematical existence is, according to MU, asserting some kind of ontological existence--no matter how many times and in how many ways we explain that this is not what anyone actually means by mathematical existence. More broadly, my impression is that MU--as the name suggests--is a philosophical foundationalist who begins with metaphysics, deriving everything else from that. Accordingly, MU has been quite dismissive of pragmatism on various occasions.

By contrast, as I said before, Peirce was a philosophical non-foundationalist, and he had harsh words for the dogmatic metaphysicians of his day. He classified the sciences in accordance with their nature and purpose, such that the more basic ones furnish principles to those above them. Mathematics comes first, because its subject matter consists entirely of hypotheses that may or may not have any basis in reality. Every other science thus relies on mathematics to some extent, not as a foundation but as a necessary tool. The first positive science is phenomenology, which deals only with what appears to the mind and identifies three irreducible elements--quality, reaction, and mediation. Then come the normative sciences of esthetics (feeling), ethics (action), and logic (thought), although Peirce generalized the last of these to semeiotic--the theory of all kinds of signs, not just symbols. Only then--after we have established the proper method for discerning truth from falsehood--do we reach metaphysics, the science of reality.

Metaphysics consists in the results of the absolute acceptance of logical principles not merely as regulatively valid, but as truths of being ... Just as the logical verb with its signification reappears in metaphysics as a quality, an ens having a nature as its mode of being, and as a logical individual subject reappears in metaphysics as a thing, an ens having existence as its mode of being, so the logical reason, or premiss, reappears in metaphysics as a reason, an ens having a reality, consisting in a ruling both of the outward and of the inward world, as its mode of being. The being of the quality lies wholly in itself, the being of the thing lies in opposition to other things, the being of the reason lies in its bringing qualities and things together. — Peirce, c. 1896

We adopt the axioms that give us a good theory. Is that Peircean pragmatism? — fishfry

As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation.

Many people who come to math through foundations believe math is about the foundations. It's the other way 'round. Mathematics comes first and foundations are just our halting and historically contingent attempts to formalize accepted mathematical practice. — fishfry

The example of synthetic differential geometry was given to show that the point of alternative foundations is to shed light on problems, not to brag about which foundation is more fundamental. — fishfry

I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not.

Thanks for the insights and the link.

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry

Just a guess, but I would imagine that one typically becomes a constructivist in the first place for primarily philosophical reasons--e.g., dissatisfaction with the philosophical basis of standard math, hence the desire for and advocacy of alternative definitions. Since this is a philosophical forum, rather than a mathematical forum, it is a natural place for committed constructivists to make their case.

"true continuity" can be defined even using standard set theory. — Mephist

No, the whole point of talking about "true continuity" is to distinguish it from (analytical) "continuity" as defined in accordance with standard set theory. The real numbers do not possess true continuity, because numbers of any kind are intrinsically discrete. However, they serve as a useful model of continuity, adequate for most mathematical and practical purposes.

Actually, even category theory can (and usually is) be based on standard set theory. — Mephist

I am not a mathematician, but my understanding is that this is exactly backwards. Set theory can be established within category theory, but category theory cannot be established within set theory. "Set" is one of the categories, but there are others that need not and do not conform to standard set theory.

But isn't he conflating human experience with reality? — fishfry

The only reality that we can know is what we learn from experience. We formulate hypotheses to explain our experience (retroduction), work out their necessary consequences and make predictions accordingly (deduction), then test whether those predictions are corroborated or falsified by subsequent experience (induction).

What is reality? Perhaps there isn't any such thing at all. As I have repeatedly insisted, it is but a retroduction, a working hypothesis which we try, our one desperate forlorn hope of knowing anything. — Peirce, 1898

Here is one especially succinct argumentation from Peirce.

We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant. — Peirce, c. 1895

It then follows from the first sentence that since the present is not an instant, there is no such thing as an instant at all.

Here is another passage that I found very enlightening when I first came across it in an unpublished manuscript.

Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible. — Peirce, 1906

Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively.

Do you know of any current mathematical objects that behave more like true continuity in your view? — fdrake

Line figures, surfaces, and solids can be understood in geometry as truly continuous. We use points to model and analyze them, but they are not composed of points.

More fundamentally, my understanding is that category theory is broader than set theory and can serve as a basis for alternative approaches to mathematics that recognize true continuity. The one that currently seems to come closest to Peirce's views is synthetic differential geometry, also known as smooth infinitesimal analysis.

Did Aristotle reject the notion of an instant of time? Or did Peirce? — fishfry

Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line.

Let's say I have some "True continuity" X. Like a line X=(0,1). — fdrake

Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line.

So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. — fdrake

No, again, a line is not composed of points corresponding to numbers. We can only mark them on (not in) a truly continuous line. They then serve as arbitrary and artificial limits/connections between distinct portions/parts.

If not, do you reject sets as a concept? — fdrake

Not at all; again, set theory can be quite useful as the basis for an approximate model of continuity. However, it cannot serve as the basis for true continuity, because it requires discreteness at the outset.