Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. — Metaphysician Undercover

Again, where is the problem if that "something" is mathematical--i.e., hypothetical--rather than actual? Are you claiming that reality is limited to that which is specific, definite, and measurable? If so, on what grounds?

A color is a quality, so its mode of being is that of possibility. Between any two "measurable" shades of red (for example)--e.g., identified by RGB hexidecimal code or electromagnetic wavelength to an arbitrary degree of precision--there are intermediate shades beyond all multitude. All of them are real, regardless of whether they ever exist by being instantiated in actual concrete particulars.

But a line is specifically one dimensional, and an infinitesimal is not. — Metaphysician Undercover

I thought it was obvious in context that I was talking about a one-dimensional infinitesimal for the sake of conceptual simplicity. Its "length" is non-zero, yet smaller than any assignable value. As such, how could we measure it, even in principle?

Respectfully, wouldn't it make a whole lot more sense if God created the universe by reason? By logic? Which is incidentally exactly how the universe functions? According to laws and relations? — Tzeentch

"Logic" is an English word derived from λόγος that names the science of symbols--signs (including words) that represent their objects only by virtue of a habit. Charles Sanders Peirce generalized it to semeiotic, the science of all kinds of signs--including indices that are directly connected with their objects, and icons that merely resemble their objects.

As I noted previously, he held that the universe itself is a sign uttered by God, whose object is God Himself and whose interpretant is our knowledge of Him. Laws and relations within the universe can also be conceptualized as signs--just like an argument, the propositions within that argument, and the terms within those propositions are all signs. So perhaps our views on this are not so different after all.

That's exactly the problem with infinitesimals, their dimensionality is ambiguous. If it is not a discrete unit in any way, then it has no form and therefore no specific dimensionality. — Metaphysician Undercover

Why is that necessarily a problem? An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality.

The main idea that I keep trying to emphasize is that a true continuum is not a composition of discrete units of any kind; it is a top-down concept, not bottom-up. Instead of division, perhaps a more perspicuous approach is to think in terms of magnification. No matter how much we were to "zoom in" on any portion of a truly continuous line, what we would always "see" is a continuous line, rather than a point or other discrete unit. Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit.

Infinitesimal points were introduced to allow that a multitude of infinitesimals may have dimensionality, therefore a continuous line could be conceived of as being composed of infinitesimal points. — Metaphysician Undercover

"Infinitesimal point" is self-contradictory--points are, by definition, dimensionless and indivisible; infinitesimals are, by definition, dimensional and potentially divisible without limit. As I have stated repeatedly, in my view an infinitesimal is not a discrete unit of any kind, and a continuum is not composed of infintesimals.

Isn't that sufficient for a continuum model? — TheMadFool

Yes, as I have acknowledged, the real numbers serve as a sufficient model of a continuum for almost all purposes within mathematics; but it is still a model, not a true continuum itself.

For me, "Word" makes sense as a translation because John goes on to say "in the beginning" in verse 2 and then "All things were made by him" in verse 3. How did God create the universe according to Genesis 1? By speaking.

You seem to have little understanding of what "infinitesimals" refers to. — Metaphysician Undercover

Right back at you.

If these continuities, space, time, and matter, could be conceived of as composed of infinitesimals (units, like monads) we could establish compatibility between a continuum and the numbers. — Metaphysician Undercover

As I already stated, a continuum is not composed of infinitesimals. Moreover, there is no ultimate compatibility between a continuum and the numbers--or anything else discrete.

So the question is whether these basic, primitive intuitions which hold space, time, and matter as continuous are correct, or are these things which appear to be continuous, more appropriately represented by the discrete units, infinitesimals. — Metaphysician Undercover

The only reason for positing infinitesimals (in contrast to points) is to preserve those basic, primitive intuitions of continuity, rather than resorting to (wrongly) treating a continuum as if it were composed of discrete units.

... this could mean that the universe is reason made physical, and that reason resides in mankind. — Tzeentch

Not really, since verses 15 and 30 unambiguously indicate that verse 14 is referring to one particular man whom John the Baptist specifically identified; namely, Jesus.

On the other hand, I agree with Charles Sanders Peirce that our existing universe as a whole is a sign uttered by God, whose object is God Himself and whose interpretant is our knowledge of Him. As the λόγος made σὰρξ, Christ is the supreme self-revelation of God (cf. Hebrews 1:1-3).

... how does one substantiate the translation of the word λόγος as 'the Word', and as referring to Christ? — Tzeentch

I am not qualified to comment on the translation, but taking λόγος as referring to Christ comes from verse 14: "And the λόγος was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth."

Yes, infinitesimals are units, they are bounded necessarily in order to give them the status of distinct things "infinitesimals". — Metaphysician Undercover

No, infinitesimals are not units, and they are not "distinct things."

However, it is necessary that either the boundaries are real in order that the infinitesimals are real, or else the boundaries are not real, in which case neither are the infinitesimals. — Metaphysician Undercover

False, infinitesimals are real but indefinite--i.e., potential not actual.

Right, but doing this only gives "potential" a different definition, just like mathematics gives "infinite" a different definition. — Metaphysician Undercover

How so? By "potential" I simply mean real possibility, rather than actuality. As Peirce put it, "the word 'potential' means indeterminate yet capable of determination in any special case" (CP 6.185; 1898, emphasis in original).

Infinitesimals does not resolve the problem because infinitesimals are units. So to model a continuum as infinitesimals is to model it as composed of discrete units. — Metaphysician Undercover

There are two fundamental mistakes here: first, infinitesimals are not units; second, a continuum is not composed of infinitesimals.

What we know, study, and understand, are all forms and forms are by definition actualities. — Metaphysician Undercover

Forms are only actualities when and where they are instantiated in concrete particulars. In themselves, as distinct from Matter, they are real possibilities.

Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". — Metaphysician Undercover

It violates the laws of classical (bivalent) logic, but that is not the only kind of logic available to us. For example, we can reason without the law of excluded middle using intuitionist logic. In fact, the law of excluded middle does not apply to infinitesimals; rather than discrete points, they are analogous to indefinite "neighborhoods" with an inexhaustible supply of potential points.

Reality and existence are not synonyms. Harry Potter and unicorns are not real, because their properties depend upon the thinking of an individual mind (e.g., J. K. Rowling) or a finite group of minds. Rocks and crocodiles are real, because their properties do not have any such dependence.

How to characterize mathematics is not as straightforward. I endorse Charles Sanders Peirce's definition of it, which he adapted from that of his father Benjamin, one of the foremost mathematicians of the 19th century: the science which draws necessary conclusions about hypothetical states of things. This clearly rules out mathematical Platonism, which claims that objects such as numbers exist; they are (logical) possibilities, rather than actualities. Nevertheless, it may still be the case that those objects are real, in the sense that we discover their properties, rather than inventing them.

Potential infinity I will leave to one side. — Devans99

Then why are we still having this conversation? I have never been arguing for actual infinity.

You can view actual infinity as the set of natural numbers { 1, 2, 3, 4, ... }. — Devans99

If the set of natural numbers is actual, then where can I find it? Again, numbers are real, but not actual.

If it's not completely defined, it's not defined at all and it does not exist logically or mathematically. — Devans99

Again, says who? Not any actual logician or mathematician, I suspect.

But there are only finite numbers — Devans99

Says who? Not any actual mathematician (pun intended).

If actual infinity is a number, there must be a number larger than any given number — Devans99

First of all, who said anything about "actual infinity"? Secondly, it begs the question to insist up-front that "there must be a number larger than any given number"; that is not how numbers work.

I'm trying to stay on the scientific side by avoiding magic. — Devans99

No, you are obstinately ignoring the difference between actual impossibility and logical impossibility. I suggest that you study up on that distinction, since it is quite important in philosophy, as this discussion has shown.

The number of events in an infinite regress is > any number — Devans99

Literally no one agrees with that statement. We could correct it to say instead, "The number of events in an infinite regress is greater than any finite number," but that is not self-contradictory at all; in fact, it is trivially true.

So just because we can imagine an mathematical infinite regress; it does[n't?] exist mathematically. — Devans99

Who said anything about "existing mathematically"? Because we can imagine an infinite regress, it is logically possible, even though it is actually impossible. See, there is that important distinction again!

A magician pulling a rabbit from a hat without using a trick of some sort. — Devans99

That is actually impossible, but not logically impossible.

Well the square is similar to a circle but an octagon is more similar but I can construct an octagon so maybe I can get there. So superficially it seems possible — Devans99

Similar ... more similar ... maybe ... superficially ... but I asked you to tell me exactly how you can imagine squaring the circle. You cannot, precisely because it is logically impossible.

An infinite regress in time for example as demonstrated earlier cannot exist in reality. — Devans99

Again, that is indeed actually impossible; but it is not logically impossible, because we can imagine it.

But just because we can imagine something it does not mean it is possible. — Devans99

Please give me an example of something that you can imagine, yet is logically (not just actually) impossible.

I can imagine squaring the circle all I like but its still impossible. — Devans99

Please tell me exactly how you can imagine squaring the circle.

A continuous circle is only possible in the mind. — Devans99

A real circle is truly continuous, and its mere possibility is sufficient for its reality.

In reality circles are made of molecules of material which are discrete all the way down. — Devans99

You mean in actuality, which is only one subset of reality. Real circles are not "made of" anything.

Have you read up on the hyperreals before? — MindForged

I suspect that I have encountered them, but it was a while back and I never got very far. I am aware of a few different approaches that seek to capture a true continuum mathematically, such as nonstandard analysis and smooth infinitesimal analysis. Again, I concede that the real numbers are an adequate model for almost all mathematical purposes.

The cardinality of the set of natural numbers is actually infinite — MindForged

Like Peirce, I prefer to say that it is really infinite, but not actually infinite. I also join Peirce in denying that numbers exist--i.e., I am not a mathematical Platonist--even while affirming that they are real.

The point of the "paradox" (not an actual paradox, just a strange thought experiment) is to point out that infinity is weird and does not work the way most people could naturally understand without learning some of the mathematical logic underlying our theories of infinity. — MindForged

Indeed, and that is the precisely point that you and I have been trying to make throughout this thread, albeit from somewhat different perspectives. Trying to treat infinity/continuity no differently from finite/discrete quantities is what leads to severe misunderstandings.

Aleph-null is larger than any natural number certainly, but aleph-null is smaller than the size of the continuum. Cantor proved this fact with a proof by contradiction (the Diagonal Argument). — MindForged

Right, and Peirce proved that the power of the set of all subsets of a given set is always greater than the power of the original set itself--which entails that there is no largest multitude (his term for aleph). What he called a true continuum is "supermultitudinous," larger than any multitude, and thus impossible to construct from (or divide into) discrete elements. You might find this introductory article about "Peirce's Place in Mathematics" interesting.

Draw a circle. There are a finite number of points on the paper as a result. There is nothing infinite about it. — Devans99

What we can draw on paper is a representation of a circle, and we can mark as many points on it as we like--up to any finite number. However, a real circle--note, not an actual circle, since there is no such thing--does not consist of any number of discrete points, finite or infinite; it is a continuous curve, infinitely divisible into smaller and smaller arcs.

just for some mathematical clarity - between any two points- whether it is on a circle or a line are an infinite number of points. — Rank Amateur

There are potential points beyond all multitude between any two actual points that we mark, but a truly continuous circle or line does not consist of points at all.

As I recall the church was somewhat split on the issue of whether God was inside or outside of time. — Devans99

Any classical theist who embraces unmoved mover arguments would "locate" God outside of time, since time would be an aspect of the universe that He created.

No I mean some things can exist in our mind and not in reality — Devans99

Right, we call such things fictions. Unicorns are not real because they are as they are only because people think of them that way.

Illogical things like inanimate objects that talk. Or infinity. Or a true continua. — Devans99

There is nothing illogical about infinity or true continua, and the fact that they cannot exist (be actual) does not entail that they cannot be real.

I think cause and effect only applying to things in time makes sense. — Devans99

In that case, your previous objections to the unmoved mover dissolve, since it is outside of time and therefore not subject to cause and effect.

Continua are real in our minds only. — Devans99

But that is a contradictory statement--"real" means "as it is regardless of what anyone thinks about it," so anything that is "in our minds only" cannot be real.

So we need a God to explain the evidence of design away. — Devans99

Design implies intention and choice from among multiple viable options, but causal determinism rules both of those out. Somehow excluding God from causal determinism would be just as "inconsistent" (according to your own assessment) as excluding an unmoved mover from being an effect. In other words, your updated axiom that all events are caused by other events entails that there is no God.

Besides, under materialism, "God" would have to be something entirely material, and thus not suitable for the name as commonly employed.

But we have larges amounts of evidence supporting the statement 'reality is modelled by maths'. If sound maths cannot represent a continuum, then thats strong evidence against continuums existing. — Devans99

Mathematical modeling is representation for a particular purpose. I already acknowledged that the real numbers serve as an adequate model of a continuum for almost all uses within mathematics, even though they do not themselves constitute a true continuum. Likewise, the mathematical models that I routinely create as part of my job are extremely useful within my professional field of structural engineering, even though they obviously are not actual structures being subjected to gravity, wind, earthquakes, etc.

And again, no one is arguing that true continua exist or are actual, but that they are real--they are as they are, regardless of what anyone thinks about them. Thus defined, reality is not coextensive with existence/actuality--there are also real qualities/possibilities and real habits/laws (conditional necessities); but that obviously takes us far beyond the thread topic.

No one is arguing for actual infinity, or that infinity is a number, or (for that matter) that infinitesimals are numbers. Again, numbers are intrinsically discrete, and thus useful for marking and measuring. On the other hand, switching from mathematics to philosophy, a real general is a true continuum--between any two actual instantiations, there are potential instantiations beyond all multitude.

Science needs to provide more empirical evidence. — Devans99

What would qualify as "empirical evidence" that the universe is cyclical, repeating the exact same sequence of events over and over? Why would we expect to find any such evidence at all?

Why then does science not take it seriously? — Devans99

Because science is causally determined not to take it seriously. In any case, a cyclical and causally determinate universe does not call for an explanation of anything, scientific or otherwise; everything is a brute fact.

I am a materialist. I see no evidence of the non-material whatsoever. — Devans99

Then why worry about finding a place for God in the picture?

No random variation in my view. No stochastic processes. Its all cause and effect. — Devans99

In that case, you must believe that this entire conversation--along with everything else that ever has happened and ever will happen--could not have played out any differently. In fact, you believe that it has happened infinitely many times in the past, and will happen again infinitely many times in the future, as the universe continues its endless (and meaningless) cycles. We are all mere cogs in a vast machine. Nothing that we observe demands an explanation; everything is just a brute fact.

How exactly do you implement transmigration of the soul? — Devans99

I see no reason why this should be problematic for the One who created the soul and the body (and put them together) in the first place, especially if we understand Him to be omniscient and omnipotent. Immense complexity does not entail logical impossibility.

I agree that the real numbers are adequate for modeling, marking, and measuring discrete quantities, no matter how small. My understanding is that conceptualizing a true continuum requires the acceptance of infinitesimals. Again, for Peirce this was primarily a philosophical matter--related to his self-described "extreme scholastic realism" regarding generals--rather than a mathematical one.

I think there might be follow on arguments that re-enforce the view that there is a God ... So if time is circular, it suggests God did it somehow (a timeless being therefore beyond cause and effect, creates another timeless entity, the universe). — Devans99

If the universe is an eternally repeating cycle, perhaps with some random variations here and there, then no timeless being is required to create it; again, existence is simply a meaningless brute fact that does not call for a rational explanation. The reality of God is much more compatible with unmoved mover arguments.

An eternal circle is the Occam's Razor design for eternal life. If I were God, it's the one I'd go for. It's actually achievable (unlike Heaven and Hell). — Devans99

Why do you think that heaven and hell are not achievable?

So I acknowledge in advance that two of my favourite ideas (Circular time and God) are hard to square together... — Devans99

Sort of like a square circle? :grin:

I think the question is whether the Planck time is properly described as a discrete "unit of time" or as a limitation on our ability to mark and measure time, which in itself is truly continuous. Needless to say, I lean toward the latter.

Cause and effect I think is a sound axiom from the standpoint of everyday experience. — Devans99

Sure, but we are discussing the nature and origin of the entire existing universe, not everyday experience. Moreover, many philosophers (and people in general) reject causal determinism, instead affirming agent causation as part of our everyday experience. For example, I am the cause of this particular post; it was not completely dictated by prior events. I could have chosen to say something different, or not to say anything at all. I could have submitted it earlier or later than I actually did.

If the universe is cyclical and governed entirely by causal determination, then the exact same series of events transpires over and over again. Even if you allow for some random fluctuations with each iteration, this is not a rational explanation of anything; it ultimately treats everything that exists and every event that occurs as a meaningless brute fact.

Also, I think I'd like to change my original axiom at this point to 'events are caused by events'. — Devans99

You have really been operating with this axiom all along, as the necessary conclusion from your two previous premises, "every effect has a cause" and "everything is an effect." Obviously no proponent of unmoved mover arguments would agree with your revised axiom, and neither would anyone who affirms agent causation rather than causal determinism. What is the warrant for presupposing that all events are caused by other events?

We do the division x=2 seconds ÷ NOW. As NOW approaches zero (the real NOW=0) x approaches infinity. That is the correct answer because between any two points in a continuous model of time there should be infinite points or NOWS. — TheMadFool

Alternatively, as Peirce argued, there are no instants (NOWs) in any continuous interval of time, and there are no points in any continuous segment of a line. Time does not consist of instants and space does not consist of points; instead, those are arbitrary discontinuities that we mark within continuous space-time for our own purposes, such as measurement. The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum.

Peirce was writing in the exact time that Cantor and Dedekind's work on infinity was contentious. It's just dishonest to pretend that has any bearing in the status of that work among philosophers and mathematicians today. — MindForged

Not sure if this was directed at me, but to clarify--I did not mean to imply that Peirce's objections to Cantor and Dedekind carry much weight among philosophers and mathematicians today. They do not, which I happen to think is unfortunate, but only from the standpoint of understanding true continuity as utterly incompatible with discrete mathematics. I readily acknowledge that treating the real numbers as if they constituted a continuum works just fine for most practical purposes within mathematics.

Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity.

The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. — MindForged

I would add Peirce here, although as in the case of philosophy, unfortunately his contributions are widely overlooked.

"Pairing", like "counting" is a human activity which is not successful unless it is completed. — Metaphysician Undercover

In mathematics, pairing and counting are not activities at all; they are concepts, and there is no requirement that they ever actually be completed, or even be capable of actually being completed. One more time: Mathematics has to do with the hypothetical, not the actual.

As I said, it's a simple and clear case refusing to simply read how the terms are defined and then pretending to have discovered a problem because some colloquial definitions of some words conflict with the colloquial definitions of other words. — MindForged

Exactly. No one is disputing that some mathematical definitions of terms are inconsistent with their colloquial (or even philosophical) meanings, but that has no bearing whatsoever on whether the associated concepts are logically possible.

Understood, and I share your suspicion that a cyclical universe is not the only alternative to an infinite regress or an unmoved mover.

Attempting to be maximally charitable, I understand @Devans99 to be arguing that there are only three possibilities:

• Our existing universe is eternal (infinite regress)
• Our existing universe is created (unmoved mover)
• Our existing universe is cyclical (Big Bang/Crunch)

All three explanations are consistent with the axiom that every effect has a cause. Adopting the additional axiom that there cannot be an actually infinite series of causes and effects rules out the first option. Adopting the additional axiom that everything is an effect rules out the second option. That leaves only the third option.

Proponents of unmoved mover arguments obviously disagree with the axiom that everything is an effect, holding instead that there must be one cause that is not itself an effect brought about by some other cause. Perhaps their additional axiom, which rules out the third option, is that the existence of something rather than nothing demands an explanation; it should not merely be accepted as a brute fact. More specifically, the existence of contingent being requires the reality of necessary being.

That's exactly my argument, it's logically impossible, impossible to conceive of, just like a square circle is logically impossible. — Metaphysician Undercover

You have offered no argument for this claim, you have merely asserted it over and over; and now you have completely undermined your own position by freely acknowledging that you are not using the relevant terms in accordance with how they are carefully defined within mathematics, such that there is no logical impossibility whatsoever.