You apparently take "rational number" to mean the same thing as "number." And you apparently think that rational numbers possess some quality that those irrational thingies don't and can't have. What quality might that be? What quality can you name, identify, describe that rational numbers share but that the irrationals cannot have - other than, of course, being rational. — tim wood

That sums it up pretty well. Irrational ideas are incoherent, so there is good reason to rid our conceptions of such things. What we like in our conceptions is the quality of being rational and we are wise not to accept irrational ones conceptions.

And if you allow for there being such a thing as the square root of two, never mind what it might be beyond being the square root of two, then you're obliged to acknowledge that it has a lot of brothers and sisters, in fact a very large family. So large that in comparison, the rationals are next to just no size at all. — tim wood

No, I do not allow that there is such a thing as the square root of two, that's the whole point, and it is what I just explained to you in the last post. Saying "the square root of two" is really a matter of saying something, which represents nothing real, just like "pi" is a matter of saying something which represents nothing real. The mathematics clearly demonstrates that there is nothing real represented by these expressions. So it really doesn't make sense to say that there is "such a thing as the square root of two", just like it doesn't make sense to say that there is such a thing as pi. These are symbols which we use because they are extremely useful, but we ought to respect the fact that there really is nothing which is represented by them. You might imagine an ideal square with a diagonal line bisecting it, or an ideal circle with a line bisecting it through the middle, but the mathematics demonstrates that these are incoherent images.

Perhaps it might help if you tell us what you suppose a number, or number itself, to be. No doubt the difficulties here originate in the foundations of your thinking. — tim wood

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? We can speak of an indefinite quantity as a quantity to be measured, when we assume that the thing is measurable, but to say that a quantity has a number is to say that it has been measured and is no longer indefinite. To say that a quantity has a number, but that the number is indefinite, is contradiction pure and simple.

Are you saying there is no square root of two? — tim wood

There is no rational number which is equivalent to what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it.

Now back to your quote. Of course we measure mathemtical objects. A unit square has side 1, area 1, and its diagonal ... well you know what its diagonal is. In fact it falls out of the Euclidean distance formula as the distance between the origin (0,0)(0,0)and (1,1)(1,1). — fishfry

OK, so we agree that if so-called "mathematical objects" are things which can be measured, Euclidian geometry creates distances which cannot be measured by that system. That agreement is a good starting point.

As a philosopher, doesn't the question, or wonderment, occur to you, of why we would create a geometrical system which does such a thing? That geometrical system is causing us problems, inability to measure things, by creating distances which it cannot measure.

No, I'm pointing out that there only seems to be a difference depending on what age one lives in. If you live in the age of integers you don't believe in rationals. You're stuck in the age of rationals and don't believe in irrationals. Matter of history and psychology.

…

We're in agreement then. But that's what a mathematical object is. A made-up gadget that, by virtue of repetition, gains mindshare. — fishfry

Maybe we can take this as another point of agreement. A "mathematical object" is nothing other than what you called a "funny gadget". Let's simplify this and call it a "mental tool". Do you agree that tools are not judged for truth or falsity, they are judged as "good" in relation to many different things like usefulness and efficiency, and they are judged as "bad" in relation to many different things, including the problems which they create. So a "good" tool might be very useful and efficient, but it might still be "bad" according to other concerns, accidental issues, or side effects. Bad is not necessarily the opposite of good, because these two may be determined according to different criteria.

Let's look at the Euclidian geometry now. In relation to the fact that this system produces distances which cannot be measured within the system, can we say that it is bad, despite the fact that it is good in many ways? How should we proceed to rid ourselves of this badness? Should we produce another system, designed to measure these distances, which would necessarily be incompatible with the first system? Having two incompatible systems is another form of badness. Why not just redesign the first system to get rid of that initial badness, instead of creating another form of badness, and layering it on top of the initial badness, in an attempt to compensate for that badness? Two bads do not produce a good.

I've studied set theory and read a number of set theory texts. I've never read or heard of any such thing. Set theory in fact is the study of whatever obeys the axioms for sets. If you ask a set theorist what a set is, they'll say they have no idea; only that it's something that obeys whatever axiom system you're studying.

You're making stuff up to fill in gaps in your mathematical knowledge. Set theory doesn't assume anything at all. It doesn't assume it's "about" anything other than sets; which are things that obey some collections of set-theoretic axioms. — fishfry

Come on, get real fishfry. Check Wikipedia on set theory, the first sentence states that it deals with collections of "objects". Then it goes on and on discussing how set theory deals with objects. Clearly set theory assumes the existence of objects, if it deals with collections of objects.

This is why it is so frustrating having a conversation with you. You are inclined to deny the obvious, common knowledge, because that is what is required to support your position. In the other thread, you consistently denied the difference between "equality" and "identity", day after day, week after week, despite me repeatedly explaining the difference to you.

But I already did. From the naturals to the integers to the rationals to the reals to the complex numbers to the quaternions and beyond. At each stage people didn't believe in the new kinds of number and though it was only a kind of "calculating device." Then over time the funny numbers became accepted. This is a very well known aspect of math history. — fishfry

You have not explained how acceptance of a mathematical tool, through convention, converts it from a funny gadget, to an object. If you cannot demonstrate this conversion, then either the tool is always an object, or never an object. Then an extremely bad tool is just as much an object as an extremely good tool, and acceptance through convention is irrelevant to the question of whether the mental tool is an object.

But do you mean how can sets be used to model mathematical objects like numbers, functions, matrices, topological spaces, and the like? Easy. We can model the natural numbers in set theory via the axiom of infinity. Then we make the integers out of the natural, the rationals out of pairs of integers, the reals out of Cauchy sequences of reals, the complex numbers out of pairs of reals, and so forth. If you grant me the empty set and the rules of ZF I can build up the whole thing one step at a time. — fishfry

Until you recognize that an "element", or "member" of a set is an "object", you are simply in denial of the truth, denying fundamental brute facts because they are contrary to the position you are trying to justify.

You haven't made any such case. — fishfry

The case I made is very clear, so let me restate it concisely. You appear to agree with me that mathematical tools are not objects, they are "mind" gadgets, yet you defend set theory which treats them as objects.

No, I took pains to make a distinction. Driving laws are completely arbitrary. But many mathematical ideas are forced on us somehow, such as the fact that 5 is prime. — fishfry

Mathematical ideas such as "5 is prime" are forced on us by the rules (laws) of the mathematical system, the definition of "prime" and "5" with deductive logic. So there is no difference. We create mathematical rules arbitrarily, as they are required for our purposes, just like we create driving laws arbitrarily as required for our purposes.

Here's an example. Take 1/3 = .333333.... Would you say that 1/3 is not resolved or requires an infinite amount of information? But it doesn't. I could just as easily say, "A decimal point followed by all 3's." That completely characterizes the decimal representation of 1/3. I don't have to physically be able to carry out the entire computation. It's sufficient that I can produce, via an algorithm, as many decimal digits as you challenge me to. — fishfry

This is nonsense. I can very easily say "the highest number". Just because I say it doesn't mean that what I've said "completely characterizes" it. We can say all sorts of things, including contradiction. Saying something doesn't completely characterize it.

It has been completely resolved. — fishfry

Unjustified, and false assertion.

It has been completely resolved. You can't write down infinitely many digits any more than you can write down all the digits of .333... But in the case of 1/3, there's a finite-length description that tells you how to get as many digits as you want. And with square root of 2, there is ALSO such a finite-length description. Would you like me to post one? — fishfry

if you switch to a different number system, one which is incompatible with the first from which the irrational number is derived, like switching from rational numbers to real numbers, this does not qualify as a resolution, if the two systems remain incompatible.

For instance, if there is infinite rational numbers between any two rational numbers, and we take another number system which uses infinitesimals or some such thing to limit that infinity, we cannot claim to have resolved the problem. The problem remains as the inconsistency between "infinite" in the rational system, and "infinitesimal" in the proposed system.

I really hope you'll consider the example of 1/3 and the fact that we can predict or determine every single one of its decimal digits with a FINITE description, even though there are infinitely many digits. Square root of 2 and pi are exactly the same. They are computable real numbers. There is a finite-length Turing machine that cranks out their digits. — fishfry

This has no relevant significance. To say "the square root of two", or "the ratio of the circumference of a circle to its diameter" is to give a 'finite description". We've already had the "finite description" for thousands of years. And, this finite description determines that the decimal digits will follow a specific order, just like your example of 1/3 determines .333.... The issue is that there is no number which corresponds to the finite description, as is implied by the infinite procedure required to determine that number.

So my analogy of "the highest number" is very relevant indeed. Highest number is a "finite description". And, the specific order by which the digits will be "computed" is predetermined. However, there is no number which matches that description, "highest number", just like there is no number which matches the description of "the square root of two", or "the ratio of the circumference of a circle to its diameter", or even "one third".

This demonstrates that there is a problem we have with dividing magnitudes, which has not yet been resolved.

It's just a representation, imperfect from the start. — fishfry

Let me return your attention to this remark. If you agree with me, that the representations are "imperfect" from the start, then why not agree that we ought to revisit those representations. Constructing layer after layer of complex systems, with the goal of covering over those imperfections, doing something bad to cover up an existing bad, is not a solution.

For the side of the square, take a straightedge, lay it along the side and mark the straightedge at the ends of the side. That's the length of the side — tim wood

The issue though, is that we are talking about measuring the ideal square, just like set theory talks about measuring the ideal numbers. We are not talking about measuring a representation of the ideal square, which is written on the paper, just like set theory is not concerned with measuring the numerals, it is concerned with "the numbers" represented by the numerals.

As pointed out repeatedly by others, the problem with irrationals is not with their number, but with some of their numeric representations. — tim wood

This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. How it is proven that the square root of two is irrational is a demonstration that it breaks the rules of "rational numbers". Pythagoras proved that the square root of two is not a rational number. This means that if we're using the rational number system, the square root of two falls outside of that system, there is no number for it. In the other thread, fishfry called this "a hole" in the rational numbers, but I disagreed with that term.

In reality therefore, an irrational number, has a numeric representation, we clearly have a numeric representation of pi, and square root two, but there is no corresponding number for these representations. As fishfry indicated in the other thread, we might create a new number system (the real numbers) and try to include what is represented by irrationals, within that system, as numbers. I do not understand the construction of the real numbers, but I am willing to argue that this is a flawed approach. Instead of addressing the real problem, which is the fact that we can produce spatial representations (circles and squares) where numbers do not apply, and adjusting these representations accordingly, the mathematicians have created an extremely complex number system, which simply veils this problem.. In other words, instead of addressing the real problem, which is a feature of our faulty spatial representations, and trying to solve that problem, the mathematicians have just hidden it under layers of complexities.

But your stance here is literally pre-Pythagorean. The Pythagoreans threw some guy overboard for making the discovery, but they accepted the fact of the irrationality of 2–√2. You refuse even to do that. You're entitled to your own ideas, but to me that is philosophical nihilism. To reject literally everything about the modern world that stems from the Pythagorean theorem. You must either live in a cave; or else not live at all according to your beliefs. You must reject all of modern physics, all of modern science and technology. You can't use a computer or any digital media. You are back to the stone age without the use of simple algebraic numbers like 2–√2. — fishfry

I accept the fact that the square root of two is irrational. That's not the issue. And I actually use the Pythagorean theorem on a regular basis working in construction, laying out foundations. The issue is that I am inclined to ask why is it the case that the square root of two is irrational, and in doing this I need to consider what it means for a number to be irrational.

To simply say as you are saying, that some numbers are rational and some numbers are irrational, and that's a brute fact, does not express an understanding of what a "number" is. But then, to ask why is it that some numbers have the property of being rational and other numbers have the property of being irrational requires asking what it means to be an irrational ratio, and one might be faced with the prospect that what we call an "irrational number" ought not even be called a "number". Perhaps the Pythagoreans threw the baby out with the bathwater, saying we can't resolve this problem so let's just call them all "numbers" anyway, and get on with the project.

So, let me state clearly and concisely what the situation is. We have a very simple looking problem of division which cannot be solved because there is no "number" which can represent the solution. You say, the problem can be solved, the resolution is an "irrational number", so just forget about that problem, it's not a problem at all. And, you say it's just "a fact of life" that some numbers are irrational. I say it's a simple fact that a so-called "irrational number" is not a number at all, because it's quite obvious that there is not a definite number which expresses the resolution of the irrational ratio. See, the very simple looking problem of division has not been resolved, and it is a pretense to claim that it has been resolved to an "irrational number".

No of course not. It tells me something interesting about abstract, idealized mathematical space. It tells me nothing about actual space in the world. — fishfry

If idealized mathematical space tells us nothing about space in the world, then physics has a big problem. But of course this is nonsense. That the square root of two is irrational, and that pi is irrational tells us something about idealized mathematical space, and that is that there is a problem with commensurability in idealized mathematical space. And, since idealized mathematical space is the tool by which we make measurements in real space, the problem of idealized space is simply ignored in application

There are no irrational distances in physical space for the simple reason that there are no exact distances at all that we can measure. So it's not meaningful to talk about them except in idealized terms. — fishfry

OK, let's talk about "irrational distances" in idealized terms then. Lets take a point A. Lets make a point B at a specific distance from point A, and a point C at the very same distance from point A, but in a direction at a right angle to the direction of point B. Do you agree that there is no definite distance between B and C? If you disagree then you are simply denying the fact. if you agree, then you might be inclined, like I am, to ask why this is the case. And so it appears to me, like there is a very real problem with "idealized mathematical space", making it less than ideal.

For all these reasons, 2–√2 is essentially a finite mathematical object. You're simply wrong that it "introduces infinity," because you have only seen some bad high school teaching about the real numbers. Decimal representation is only one of many ways to characterize 2–√2, and all the other ways are perfectly finite. — fishfry

You might say "√2" is a finite mathematical object, but until you define what a mathematical object is, it's you who's just typing word salad. In reality "√2" is an unresolved mathematical problem. That you call it a "mathematical object" doesn't mean that it is a "number", nor does it mean that it actually is an object. And, when one attempts to represent this so-called object as a number, infinity is introduced

However, I didn't say that it "introduces infinity" on this thread. If I mentioned that, it was another thread in another context. Perhaps I said that in a thread on infinity. What I am focused on here is simply the meaning of an irrational ratio, and whether it is appropriate to claim that the ratio has been resolved to a "number", called an "irrational number".

But to simply say that you don't like 2–√2; that's just a hopelessly naive viewpoint. — fishfry

It's not "√2" that I dislike, it's what it represents that is what I dislike. And it's not that I am simply saying this, I am giving you the reasons for my dislike. But you seem to be good at ignoring reasoning.

It's just a representation, imperfect from the start. — fishfry

Right! That's why we ought to seek a better one! That's exactly what I'm arguing. Don't you agree?

I have been reading Meta's post and it's hard to know where to start. @Meta, do you agree that math isn't physics? You complain that we can't measure 2–√2 but we can't measure 1 either. — fishfry

I am satisfied with this principle if we can apply it consistently. We do not measure mathematical "objects", they are tools by which we measure objects. That's why I argued that they are not proper "objects". Now let's apply this to set theory. Cardinality, for example is a measure. If the applicable principle is that we do not measure mathematical "objects", then why allow this in set theory? It's inconsistency.

So either we can measure mathematical objects, like squares, and the sides of squares, just like we can measure the cardinality of sets, or we cannot measure these so-called mathematical objects. I'm fine with the latter principle so long as we maintain consistency. But if we allow that we can measure these so-called objects, then we can measure a square, and find that the diagonal cannot be measured.

Yes, you've got the diagonal of a square. Why is that distance "immeasurable"? — tim wood

It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry?

I believe I did. You think math should be physics. You think perfectly accurate distances exist in the real world. You seem to deny abstraction, as in your claim that squares are "impossible," your word. — fishfry

This is not at all what I've been saying, so I think we might not really be making any progress.

In each case, mathematicians of a given historical age become interested in an equation they can't solve. They say, "Well what if there were a funny gadget that solved the equation?" Then after a few decades the made-up gadget becomes commonplace and people come to believe in it as a first-rate mathematical object, and not just a convenient fiction. Eventually we can construct these gadgets both axiomatically in terms of their desired properties; and as explicit set-theoretic constructions. They do indeed become real. Mathematically real, of course. Nobody ever claims any of this stuff is physical. You are fighting a strawman. — fishfry

Neither you nor I is talking about physical objects here. What we are talking about is the "made-up gadgets" which you describe here. You seem to imply that there is a difference between these funny gadgets, and "first-rate mathematical objects" I deny such a difference, claiming all mathematics consists of made-up gadgets, and there is no such thing as mathematical objects. But this is contrary to set theory which is based in the assumption of mathematical objects. If you really think that a "funny gadget" becomes a "mathematical object" through use, you'd have to demonstrate this process to me, to convince me that this is true.

So this is where I want to engage. You seem to want math to be what it can never be. You want an abstract, symbolic system to be real, or actual. That can never be. Your hope can never be realized. — fishfry

How can you not see that this is a problem for set theory? Set theory assumes that it is dealing with real, actual mathematical "objects". That is a fundamental premise. Now you agree with me, that mathematics can never give us this, real or actual things being represented by the symbols. So why don't you see that set theory is completely misguided?

But abstractions can be real as well. Driving on the left or right is a social convention that varies by country. It's artificial. Made up. It's nothing but a shared agreement. It could easily be different. Yet it can be fatal to violate it. Driving laws are social conventions made real. See Searle, The Construction of Social Reality. — fishfry

So your argument is that the "funny gadget" gets made into a "first-rate mathematical object" through convention, just like driving laws. But those are ";laws", not "objects". Let's suppose that the mathematical symbols referred to conventional laws instead of "objects", as this is what is implied by your statement. How would this affect set theory? Remember what I argued earlier in the thread, sometimes when a symbol like "2" or "3" is used, a different law is referred to, depending on the context.

I submit that 5 is prime and the square root of 2 both exists and is rational. — fishfry

I don't see how "the square root of 2 exists" could possibly be true, It is an irrational ratio which has never been resolved, just like pi. How can you assert that the solution to a problem which has not yet been resolved, "exists"? Isn't this just like saying that the highest number exists? But we know that there is not a highest number, we define "number" that way. Likewise, we know that pi, and the square root of two, will never be resolved, so why claim that the resolution to these problems of division "exist"?

For you to see the color, it takes you and the flowers, plus the other sensory aspects of the medium at the time you are looking at them. The color is not confined to the flower. Your seeing it is caused by many things other than you and the flower, and without that combination of all aspects (which we are still discovering) of the medium at that moment in time, you would not see the color, or perhaps that same shade as another person would. We each see color differently because of this. It is manifested by the continuum. And other life forms would see it even more differently than humans. — Mapping the Medium

How is this evidence that continuity is real?

On the assumption that there can be two points the distance between them being "measurable," how do you "very easily make two points with an immeasurable distance between them"? — tim wood

Sorry, I didn't understand your challenge, you said something obscure about making you go away. I'm afraid you have your own free will and I can't make you go away.

Now that you've made you question clear, I'll answer it. But it should be obvious from what I already wrote, so I don't understand why you're confused. Construct a square according to the rules of Euclidian geometry, the opposing corners are an immeasurable distance apart.

Colour is a fascinating subject. When I see an array of flowers, each having its own unique special blend of hues, on a summer day, I am awestruck by the beauty, and the fact that each particular colour is something created by that individual living being. But I don't see what this has to do with the reality of continuity. In fact, it seems more like evidence of the reality of individuality.

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. — aletheist

Indefinite, means unlimited, which is the same as infinite. So you've just led me around in a circle. We're back to where we started. And so I'll ask you the same question again. Do you understand the difference between "infinite" and "infinitesimal? "Infinite" implies unlimited, while "infinitesimal" implies a limit.

So all you have done now is removed the distinction between "infinite" and "infinitesimal" by claiming that infinitesimals have no boundaries. You are steeped in contradiction, trying to maintain the difference between infinite and infinitesimal, while at the same time trying to remove the very thing which constitutes that difference, the boundaries which infinitesimals necessarily have.

Ah, so now you are claiming that Peirce was either self-deluded or a liar. — aletheist

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

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. — aletheist

That's what I have been doing. The "infinitesimal" of Peirce is nothing other than prime matter as described by Aristotle. And your practise of contradiction as the only way to defend Peirce, along with your claim that Peirce allows for violation of the principle of contradiction, is indicative of dialectical materialism. Since dialectical materialism (as associated with Marxism) was not well respected in the United States, it makes sense for Peirce to offer another name for his metaphysics, objective idealism. Notice that both dialectical materialism, and objective idealism are derived from the Hegelian concept of "becoming"

There is truly continuity in all things (synechism). — Mapping the Medium

This is the unsupported premise, the thing taken for granted which no one seems to be able to back up with reasonable principles.

What we have so prevalent in our world today due to those medieval misguided turns, is the slicing and dicing (nominalism) and the missing of a hugely important component (Cartesian dualism= diadic, versus what should be triadic), ultimately encouraging the idolization of the 'individual'. — Mapping the Medium

Until real continuity can be demonstrated, the only true starting point is the "individual". The reality of the individual is supported by the existence of the "medium" which separates and distinguishes individuals, making the only acceptable starting point for any logical proceeding, the law of identity. The portrayal of the medium as creating a continuity is demonstrably a false representation, as the name "medium" implies.

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. — aletheist

All you are doing is qualifying "boundaries" with "distinct", and insisting that infinitesimals do not have "distinct boundaries". Nevertheless, infinitesimals require "boundaries", as I said, and therefore a continuity cannot be composed of infinitesimals because these boundaries necessarily break the continuity whether they are distinct or not.. Saying that the boundaries are vague and not distinct, does not say that there are no boundaries. And if there are boundaries there is no continuity.

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

Someone like Peirce, can say "I am not materialist, my metaphysics is objective idealism", and still offer us a metaphysics based in materialist principles. So I don't see how this claim is relevant.

As for forms, your comments are all over the place. For Aquinas God has infinite form, angels are form, and humans are form and matter. When a human understands something, it's form enters the intellect. That's it. There is not much else to his philosophy on this. I have no idea where you are going with your posts on here — Gregory

Do you recognize the fundamental Aristotelian principle, upheld by Aquinas, that there are two distinct types of "forms", the form of the particular (complete with accidents), and the universal form, abstracted by the human intellect? Because of this difference, it is incorrect to say that the form of a thing enters the human intellect. The form of the thing is a particular, whereas the form in the human intellect is a universal, it is an abstraction which does not contain the accidentals.

Because of this, we need to account for the process of abstraction, which is not a matter of "its form enters the intellect". This is why Aristotle introduced a division between the active (agent) intellect, and the passive intellect, a division which Aquinas upheld. The exact nature of these two, or even if the distinction is warranted, is what is at issue in the Nominalism/Realism debate.

Simply put, if the intellect receives forms, through sensation etc., or any other means, it must have a passive, receptive, aspect. This passive aspect is of the nature of potential, which is substantiated by Aristotle, as matter. This poses a somewhat vexing problem for St Thomas who wants to maintain the immateriality of the intellect. So he is inclined to posit a potentiality which is proper to the soul, but is not a material potentiality, to account for the passive intellect. The agent intellect, as pure act now, must be properly positioned as independent from the human body.

The nominalists, following an interpretation of Aristotle which was probably derived from Avicenna and Averroes, wanted an inversion of this position. They wanted the active intellect to be within the individual's soul, and the passive intellect to be external to the individual, in the realm of matter. But this appears to leave no way to validate the universal forms as they are evident, being proper to the human intellect. The nominalist must therefore demonstrate how the active intellect, within the individual human being, creates the universals, and gives them to the passive intellect in the material realm, in order to validate the existence of universals. Someone like Ockham might slice through these complexities, denying the reality of universals thereby denying the need for a passive intellect, claiming that all there is is symbols (words) in the material realm, and universals are simply words.



.

I assume you mean there are many sets of two points in which the distance between the two points in each such set is "measurable." As opposed to the two points "easily made" you mention. You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. — tim wood

You seem to be lacking in reading skills tim, it's no wonder you're so confused. I was talking about what someone can do with Euclidean geometry, not about what someone can do walking on the ground. So your example is way off track.

If you want to argue that mathematical objects and geometrical constructs are not real objects, then I'm in agreement with you there, and that's the position I've taken on this thread. And, the fact that we can create an idealized object (created through the use of defined terms, rather than drawing, or walking on the ground), like a circle or a square, and make these objects such that there is immeasurable distances within the objects, is further evidence that these are not truly "objects". My argument has been, that a mathematical object, like what is referred to by "2" or "3" does not qualify as an object in any way consistent with the law of identity. And set theory is based in, and requires, this false premise that these are objects. In other words, set theory violates the law of identity.

If you're interested, go back and read the thread, Zuhair and I covered much ground. Fishfry was in and out, and for the most part did not keep up with the discussion, but now seems to have a renewed interest.

God creates the division. Augustine and Aquinas explicitly say the forms are in God. The doctrine of Plato that the forms are separate from God are held by few Christians. — Gregory

Aquinas clearly distinguishes independent Forms ( God's intellect, and angels) from those forms dependent on the human mind (abstractions).

Aquinas supported the Inquisition. — Gregory

I think you have your information mixed up. Aquinas was prior to the Inquisition.

I've read through much if not most of your writings in this thread. I think I understand where you're coming from. I found two remarks I can get some traction on. — fishfry

Thank you fishfry. I'm vey impressed that you actually took the time to read and try to understand what I was saying. Most just dismiss me as incomprehensible or unreasonable and go on their way.

So ok. ZF[C] is unsound. This is a commonplace observation. Might you be elevating it to a status it doesn't deserve? Are you thinking of it as an endpoint of thinking? What if it's only the beginning of philosophical inquiry? — fishfry

As Aristotle explained, logic proceeds from the better known, toward understanding the lesser known. The premises are the better known, the conclusions lesser known. A conclusion requires multiple premises so if there's a possibility that each premise is incorrect, the probability is multiplied in the conclusion. Therefore the "endpoint" of thinking is more uncertain than the beginning point. The engineer applies mathematics as if there is a high degree of certainty in the axioms (premises), and uses these toward producing an understanding of what is until then unknown.

The principles we apply in any philosophical inquiry must be of the highest degree of certainty in order to give any credibility to the conclusions of the inquiry. Science and engineering apply mathematics with a confidence that the underlying axioms are sound. If the engineers thought that the mathematical axioms were unsound, they would request better ones. A philosopher such as myself may approach the axioms with skepticism. But the skepticism must be justified by underlying principles with higher certainty. If one were to cast doubt on any particular axiom, that philosopher must appeal to further principles which are known with an even higher degree of certainty. That's what I believe I am doing, referring to principles with a higher degree of certainty, such as the laws of identity and non-contradiction, to cast doubt on these mathematical axioms.

What Russell says is true to an extent, people who apply mathematics do not generally question the accuracy of the axioms. However, in the passage quoted he fails to mention the reason for this. The reason that they do not question the accuracy of the axioms is that they have confidence in those axioms. So yes, applying mathematics is no different from applying other logic, we proceed with the underlying knowledge that if the axions are true then so will be the conclusions, when the logic is properly applied. However, what is not mentioned in the passage is that when we apply mathematics we have a high degree of confidence in the truth of the premises, and that's why we are using mathematics.

That, I submit, is one of the key issues of the philosophy of math. We agree that math isn't true, but it nevertheless seems intimately related to the real or the actual.

This is what I mean when I say that "ZCF is unsound" is the starting point of philosophical inquiry into the nature of mathematics; not the end of it. I get the feeling you think it's the last word. It's only the first. — fishfry

Perhaps you and I can find an agreeable starting point for a philosophical inquiry, here. We agree that the axioms of mathematics are not true, so we can now examine the confidence with which mathematics is applied. Mathematics does not consist of "true principles", so the confidence is not based in truth. I suggest that the confidence is based in utility. It has worked in the past, it works today, therefore it will work in the future. Mathematics is reliable, therefore we have confidence in it.

Now, this reliability indicates that mathematics is, as you say "intimately related to the real or the actual". I have no problem agreeing with you on this. So we can ask, what produces this reliability, what is the nature of this relationship with the real or actual. I propose that the reliability is produced by approximating the truth. Reliability does not require that we know the absolute truth concerning the matters we are involved with, but it does require that we have an approximation of the truth, and this approximation lowers the probability of mistake, increasing reliability. The nearer we can approximate the truth, the higher will be the reliability. In this way we remove "truth" from the absolute logical categories of is/is not, placing it into the relative, degrees of probability. Propositions are not judged to have truth or falsity in an absolute way, they judged for probability of truth.

That's just silly. There's nothing fundamentally unsound. The diagonal of a unit square can't be expressed as a ratio of integers. It's just a fact. It doesn't invalidate math. Even the Pythagoreans threw someone overboard then accepted the truth. — fishfry

Didn't you just agree that in a very fundamental way, mathematical axioms are unsound? And you supported this with the quote from Russel. Or do you think it's just some axioms which are unsound? ZF is unsound but Euclidian geometry is sound. What about the parallel postulate? Anyway, if some axioms are sound and others unsound, we'd have to revisit all the axioms anyway, to distinguish the degree of soundness.

Sure, it's "a fact" that Euclidian geometry produces points which have a distance between them as immeasurable, but it's also a fact that this is a problem which has not been resolved. We've agreed not to speak in terms of soundness or truth of the axioms, so all we can look at is whether the axioms are problem free, and these geometrical premises create problems, i.e. that there are points with immeasurable distance between them.

That it comes up so naturally, as the diagonal of a unit square, shows that (at least some) irrational numbers are inevitable. Even if we're not Platonists we suspect that a Martian mathematician would discover the irrationality of 2–√2. There's a certain universality to math. Euclid's proof is as compelling today as it was millenia ago. The philosopher has to acknowledge and account for the beauty, simplicity, and power of Euclid's proof, undiminished for 2400 years; not just deny mathematics because some number doesn't happen to be rational. — fishfry

Let me apply the principle described above, to √2 now. Truth is relative, not absolute. First, it is probably not true that a square is natural. The right angle is artificial. It was produced for various purposes, surveying plots of land, establishing parallel lines, etc.. So it is highly unlikely that √2 comes up naturally, it comes up as a result of our desire to create parallel lines, or whatever other purposes we use the right angle for. Therefore, our desire to create parallel lines and such, has produce the right angle, which is extremely reliable for these purposes, so it must approximate the truth about "space" to a fairly high degree.

However, there is this problem which the right angle creates, and that is that it allows us to very easily make two points with an immeasurable distance between them. The "circle" and "pi" has a very similar problem, so this geometrical system based on a point with degrees of angles around the point is suspicious. Now, I see two ways we could go with this inquiry. We could follow principles like Peirce's and say that's simply the way space is, there's a vagueness about it which incapacitates us in this way, making it impossible to measure the distance between these points. The nature of space is such that we cannot determine points in space, only vague infinitesimals. That assigns "the fault" to the object, space. Or, we could proceed in the way that I recommend, and consider that our desire to make parallel lines and such, has misguided us relative to truth, such that we modeled space with right angles etc., which was an approximation to the truth, but maybe not accurate enough for the purposes we now have. This assigns "the fault" to the subject, the model. The problem being that the circle and the right angle do not properly represent the spatial area around the point. Notice that the latter way, which I propose, gives us inspiration to delve into our spatial models, divulge our past mistakes and produce a better approximation of space. The former way suggests that it is impossible to model space more accurately, because we cannot model a precise point. And that's just the way that space is, impossible to model in such a way as to rid ourselves of the conclusion that if locations are represented as precise points, there will be points with an immeasurable distance between them. So we are uninspired to look for that better way of modelling space. Therefore, if we are inspired to increase the degree of reliability and certainty with which we apply mathematics, we need to revisit this model of space, to see where it leads us astray.

Among the irrationals, the very simplest are the quadratic irrationals like 2–√2, meaning that they're roots of quadratic equations. They're so easily cooked up using basic algebra, or Turing machines, or continued fractions. — fishfry

That irrational numbers are "easily cooked up" from our axioms is clear evidence of weakness in the axioms. As explained, irrational numbers represent things which cannot be measured using the existing axioms. If more and more axioms need to be layered on in a seemingly endless process, to create the appearance that these immeasurable things actually can be measured, this is simply closing the barn door after the horse has run away.

The incommensurability which produces irrational numbers.

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. — aletheist

Infinitesimals are within a continuous medium, and they also require boundaries. Therefore they are artificial, according to what you say here.

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. — aletheist

Exactly as I said, Peirce allows for violation of the law of non-contradiction. Therefore he is dialetheist, and in my judgement, dialectical materialist. It is difficult to hold a "process" type metaphysics as Peirce does, without turning either to God or dialectical materialism for foundational support. Since Peirce allows for violation of non-contradiction his appeals to God are vacuous.

Again, hair-slitting. You either believe that Platonic Forms are real entities separate from God or not. Aristotle, Augustine, and Aquinas all thought not. Neo-Platonic Christianity and Christian Aristotelianism are not different. They just have a little different emphasis — Gregory

Are you sure about this? I wouldn't agree to that. Do you recognize that Aristotle identified two types of substance, primary and secondary? Substance has real existence, and also must have form. Yet there are two distinct types of substance, this is fundamental to dualism. If no forms are separate from God, then why are there two distinct types of substance? What creates that division?

Seems that way to me too. Our friend Metaphysician Undercover, who must be a neo-Pythagorean, is mightily vexed by the fact that the square root of 2 is (a) a commonplace geometric object, being the diagonal of a unit square; and (b) doesn't happen to be the ratio of any two integers.

What of it? Humans got over this about 2500 years ago. — fishfry

Human beings may have gotten over this, but they did not resolve the problem. Consider the problem this way. Take a supposed "point". Now measure a specific distance in one direction, and the same distance in a direction ninety degrees to the first. Despite the fact that you use the exact same scale of measurement, in both of these measurements, the two measurements are incommensurable. Why is that the case?

Doesn't this tell you something about the thing being measured (space)? What it tells me, is that this thing being measured (space), cannot actually be measured in this way. The irrational nature of pi tells us the very same thing. Two dimensional objects have a fundamental problem which demonstrates that space cannot actually be represented in this way.

We see a very similar problem in the relation between zero dimensional figures (points), and one dimensional figures (lines), as discussed in the other thread. So if we get done to the basics, remove dimensionality and focus solely on numbers, we can learn to understand first the properties of numbers, quantity, and order, without applying any relations to spatial features. Then we can see that it is only when we apply numbers to our dimensional concepts of space, that these problems occur. The problems result in establishing a variety of different number systems mentioned in this thread. None of these numbers systems has resolved the problem because the problem lies within the way that we model space, not within any number system. We do not have a representation of space which is compatible with numbers.

It was quite a shock to them when they came to realise that the square root of two could not be expressed as a fraction! There was no alternative except to introduce real numbers. — A Seagull

The problem though is that introducing real numbers does not actually solve the problem, it just offers a way of dealing with the problem. So the problem remains and mathematicians simply work around it with increasingly complex number systems.

So, my challenge is if someone can construct the real numbers in a concise and clear way that the average student starting calculus in high school would easily understand for then transcendental constants like pi to make perfect and clear sense and all the tricky questions above perfectly clear answers (just as clear as in geometry or proofs about discrete numbers). — boethius

So you ought to see, that there is no clear and concise way to construct the real numbers. "The real numbers" is an extremely complex way of working around a very simple problem. The problem, as explained above, is that we do not model space properly. This creates problems with applying numbers to spatial representations. So instead of addressing the real problem, which is our representation of space, mathematicians continue to create exceedingly complex number systems in an attempt to work around the problem. Of course the problem remains though, so new issues pop up, and mathematicians continue to layer on the complexity.

As philosophers, who have met the problems of mathematicians and have chosen philosophy instead, we might focus on the real problem.

Infinitesimals/moments are indefinite, not distinct. — aletheist

I know, I explained this, the boundaries between moments are vague. That's why we can throw the "point" out the window. The "point" does not apply, as special relativity demonstrates. So the point is useless and that's why I said Peirce dismisses it.

The principle of excluded middle does not apply to them. — aletheist

Claiming that the law of excluded middle may be violated does not resolve the problem. It is a matter of contradiction. The existence of infinitesimals in the medium requires that there are natural boundaries.
If there are boundaries within the medium, then the medium is not continuous. Saying that there are boundaries which are "indefinite, not distinct", and concluding therefore that they are not boundaries at all and therefore there is continuity, is simple contradiction.

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

Scotus.....

- Denies whatever is 'one' is an individual. — Mapping the Medium

Can you elaborate on this point? How is there a difference between being "one" and being an "Individual"?

- Accounts for causation in this 'degree of less than numerical'. (Experience and events provide this 'degree' of influential causation. Think about what science is now discovering about epigenetic/environmental/experiential influences.) — Mapping the Medium

And could you expound on this as well? What is this feature of reality which you call "less than numerical", what is this Unity which is less than numerical?

Neo-Platonist principles, which I believe ground Christian metaphysics holds the fundamental unity as the "One". Neo-Platonism is a sort of unified Plato/Aristotle perspective, prior to the more comprehensive unification provided by Aquinas. This prior conditioning of Neo-Platonism (Augustine for example), into Christian metaphysics, is prerequisite for Aquinas' approach towards making Aristotelian principles consistent with Christian principles, thus facilitating this process.

Therefore, I am interested in where this idea of a unity which is somehow not consistent with the arithmetical numerals is derived from. How is there such a thing as a unity which cannot be represented by the numeral "one"? What makes this "unity" different from a normal "unity" which we represent with "one"?

Singular essences are unknowable to us, even though they ARE real. We refer to their reality indirectly by recognizing and differentiating what it is not. Example: Humans develop and recognize 'self' only in relation to that which is 'not' self. — Mapping the Medium

Isn't this fundamental to the nominalist approach? And it is consistent with Aquinas as well. Based in Neo-Platonism, Aquinas emphasizes a distinction between divine Forms with independent existence, and the (abstracted) forms of the human mind, which cannot be independent from matter because they dependent on the human mind which is attached to matter.

It is Aristotle's metaphysics which explains how the Form of the particular, singular, or individual object, must be prior in time to the material existence of that particular, or individual. This necessitates the independent Forms. of the Neo-Platonists. Notice that these Forms are forms of particulars, grounded in the "One", (the Form of the entire unity, the universe). They are not Platonic forms, which are universals which depend on the human mind.

The extreme difficulty is approached in Plato's Timaeus, which is to establish a relationship between the form of the individual, and the form the universal. There are numerous approaches offered, over the centuries, and each gets tangled and lost in numerous twists and turns, and category mistakes. Plato tries to offer the straight and narrow, but puts the universal form as prior, while Aristotle seeing the dead end to this approach turns things around to place the form of the particular as first, and this is represented in Neo-Platonism as the "One".

But again, the nominalists and the realists are still both misguided. So we have all of these 'camps' of thought going round and round on this merry-go-round, and never getting off. — Mapping the Medium

I would not say that either of these camps are "misguided". They each manage to approach first principles, in their own way. To get to this point, of approaching first principles requires serious guidance. However, each has a slightly different approach, and neither manages to get beyond what I called the "extreme difficulty", so both appear to be somewhat misguided. But since the extreme difficulty remains unresolved, we cannot really say that they are misguided because the proper guidance remains unavailable. We'd have to say everyone is misguided, even though they managed to get this far.

The only way to make any difference in what has happened is to try and teach the general public how human beings actually develop and how life interacts with each other. If we only recognize ourselves and our 'medium' by what it is not, then we have to realize that the only way to learn and reach a shared understanding is through dialogue with others who have a different perspective. — Mapping the Medium

This sounds like a Wittgensteinian perspective, which is not really dissimilar to Peirce. The individual's perspective is first, the primary perspective, and we create a shared understanding. The issue is whether we can overcome differences between the individual perspectives, to validate real, true, universals, rather than taking the realist perspective, that the universal is real, prior, and imposes itself on us.

Distinct points/instants are indeed arbitrary and artificial creations of thought, but indefinite infinitesimals/moments are real, with length/duration less than any assignable value and no discernible boundaries. — aletheist

Do you not see the problem here? You yourself said "there is still a role for points--not as the parts of a line, but as the discrete boundaries between its continuous parts". An "infinitesimal" requires such a boundary to exist as an infinitesimal. Do you agree with this, there is no infinitesimal without a boundary to separate it from others? If the boundary is arbitrary, an artificial creation, as you say here, then so is the infinitesimal created by the boundary.

Where do you get the idea that Peirce thought there were infinitesimals which are not created by those arbitrary divisions? As I explained, if the boundaries required for such infinitesimals, actually existed within the medium, then these boundaries would break up the continuity of the medium, such that it would not be continuous.

Peirce does not "replace" points with infinitesimals; they are two different concepts, and there is still a role for points--not as the parts of a line, but as the discrete boundaries between its continuous parts. — aletheist

As I said, these "discrete boundaries" cannot be real, because they would break the continuity. They are arbitrary and artificial, imposed as our pragmatic divisions in the continuity of time. Nor can such boundaries be "points", because they are vague (and this is borne out by special relativity), due to a combination of our inability to impose an actual "point" into the passing of time, and there being no actual points within the passing of time. Therefore the traditional concept of "point" really has no role in Peirce's metaphysics. Though he might discuss the "point" it's something he rules out as impossible, with his principles, just like Aristotle discusses "prime matter", but rules it out as impossible, with his principles.

He helpfully clarifies this in one manuscript (R 144, c. 1900) by referring to points (or instants) as limits and the line segments (or lapses of time) between them as portions. In later writings he reverts to "parts" for the latter, but suggests "connections" for the former. — aletheist

Peirce is very clear in his principles, that such boundaries or "limits" are vague. Therefore they cannot be points according to the classical definition. That you can refer to one time when he pondered the existence of points, does not mean that he didn't dismiss them altogether at a later time. The fact that he later calls them "connections" rather than "points", is an indication of this. The "connections" between the "parts" of temporal duration are not produced by clear cut points of "now", they actually must have extension inherent within, as is indicated by what is expressed in special relativity, the impossibility of giving "now" a particular point.

No, the very nature of infinitesimals/moments is that they are not distinct from one another at all. — aletheist

Yes, sure, but this is the very point I am arguing. The infinitesimals of the continuity are not actually distinct from one another, they are only separated through the arbitrary imposition of "points". However, we as human beings have not got the capacity to insert true "points" into a natural continuity. Therefore the so-called "points" which we actually use are not really points at all, they are divisions with vagueness inherent, not points. So we can throw the concept of "point" right out the window because it has no purpose for us.

f we have good reason from our phenomenal experience to posit that continuity is real, and the hypothesis of infinitesimals "provides the logical foundation for the reality of continuity," then we have good reason to conclude that infinitesimals are likewise real. — aletheist

The problem is that we know sensations deceive. A succession of still frames creates the illusion of continuous motion. So we cannot simply assume continuity is real.

What is the argument for denying the reality of infinitesimals? — aletheist

If the passage of time, or anything else for that matter, is assumed to be continuous, then it is assumed that there are no real divisions within that continuity. If that continuity is divided into parts, such a division is done for pragmatic purposes only, the divisions are arbitrary in that sense, not based in the assumption of any real parts, or real divisions. Therefore any "infinitesimals" produced by such a division are artificial, arbitrary, having no real substance to the divisors. And, we might just say that any such infinitesimal could actually be divided again, ad infinitum. There is no substance to the infinitesimal.

Instantaneous states are creations of thought for describing real events in time. We arbitrarily mark them at finite intervals, but the reality is continuous motion/change. — aletheist

Right, and since such "instantaneous states", or what you called above, "points", are required as divisors, to create the infinitesimals. Therefore the infinitesimals themselves are just creations of thought. Then, if we add the further development to Peirce's thought, that such "points" are not points at all, but vague divisors, we find that the infinitesimals themselves are lost into a veil of vagueness. Therefore our capacity to understand the thing which appears to us as continuous motion/change, completely breaks down and is lost into this incoherent sea of vagueness, that is if we adopt these principles.

Nonsense, Peirce consistently affirms that time is (potentially, not actually) infinitely divisible, and that this is always necessary (but insufficient) for true continuity. In fact, he asserts repeatedly that instants of any multitude, or even exceeding all multitude, may be inserted within any lapse of time--even an infinitesimal moment. — aletheist

if this is the case, then it is clear evidence that "infinitesimals" are completely fictitious and serve no purpose in the understanding of continuity. But this is not what Peirce argues so I do not believe it is true.

My ears tell me sound is infinite, when I study music. — Gregory

Even your ears have physical limitations. There's no way you could distinguish an infinite number of different notes. And if you just assume that there is so many different notes that they must be infinite, that's an unsound premise.

So your argument on this thread is that there is not a contradiction in math, but that it's incomplete? — Gregory

Yes it is incomplete, but I think what I meant was that there is contradiction in math but it's acceptable because it's unavoidable. It's unavoidable because mathematics is a reflection of our understanding of reality, and our understanding of reality is limited by our capacities as human beings. So contradiction just reflects our imperfections. If mathematics were without contradiction, it would be perfect, and our understanding of reality would be perfect. The incompleteness is therefore our inability to completely rid the system of contradictions.

Most mathematicians and physicists do quite well without contemplating such issues. — jgill

I agree, these issues are simply accepted, taken for granted, perhaps almost subconsciously, and not worth thinking about for most mathematicians and physicists. They work with the tools they have. Likewise, that our knowledge of the physical universe is incomplete is also taken for granted. If we put two and two together, we might conclude that a better system would provide us with a more complete knowledge.

How much of Peirce's metaphysics (and mathematics, and phenomenology, and logic/semeiotic) have you actually studied carefully? What fundamental distinction are you positing here between the concepts of "infinity" and "infinitesimal"? — aletheist

Are you not aware of the difference between infinitesimal and infinite?

Peirce would agree with this, although "infinitesimal point" is a contradiction in terms. There are infinitesimals, and there are points; they are two very different concepts, since infinitesimals have extension (though smaller than any assignable/measurable value), while points do not. His parallel terms when discussing time are moments, which have duration (though shorter than any assignable/measurable value), and instants, which do not. — aletheist

Let me rephrase that then, Peirce replaces the "point" with the "infinitesimal", as the point might be designated as unreal, and incapable of producing a continuity. The duration of time cannot consist of "instants", or points, which have zero duration, but it may consist of "infinitesimals", which I might have carelessly referred to as points with extension.

Peirce would agree with this, as well. Infinitesimals (and moments) are indefinite, and thus cannot be individually distinguished; we can only discern differences once we have marked off specific points (or instants). — aletheist

This is why such infinitesimals cannot be taken as real. Each infinitesimal requires a point of division, a boundary, to separate it from another infinitesimal. Without such a boundary the infinitesimal has no existence. But these boundaries are said to be vague because such dimensionless points cannot have any real existence in an extended medium. So Peirce proposes nothing to substantiate any boundaries and therefore nothing substantiates any infinitesimals. The infinitesimals are imaginary, simply a proposal, as that which constitutes the continuity. But the reality of the continuity is only supported by the infinitesimals. So the position is in fact, circular.

Let me explain better. For various reasons we are inclined to assume the reality of continuity. However, the existence of change and difference makes it very difficult to validate logically any supposed continuity. If infinitesimals are real, this provides the logical foundation for the reality of continuity. But the only thing which supports the reality of the infinitesimals is the "need" to support the continuity. Of course it's a pragmatism, the infinitesimals are assumed for the purpose of making continuity real, but there is nothing real to support this "need".

In his own words, "between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false [Cantorian] continuity which the calculus recognizes but in a much stricter sense." — aletheist

See, the problem here is that the "two instantaneous states " are not real. There is nothing to validate the still "instant" in the continuous passage of time. And if such instants were real, they would break the continuity. They are only posited to allow that the infinitesimals which exist as continuous change between the instants, are real. But if divisions in time are created artificially by positing such points, then there is no principle to deny dividing time infinitely. So the infinitesimals are posited solely for the purpose of denying infinite division, without any real substance.

Peirce would vehemently deny both charges here--he does start with a pure and true continuity as his first principle, or at least consistently strives to do so; and he explicitly rejects materialism, calling it "quite as repugnant to scientific logic as to common sense," instead affirming objective idealism as "the one intelligible theory of the universe." It treats "the physical law as derived and special, the psychical law alone as primordial," such that "matter is effete mind, inveterate habits becoming physical laws." Accordingly, Peirce's cosmology understands the very constitution of being as true continuity underlying indefinite possibilities, some of which are actualized by the ongoing process of determination. — aletheist

It's very clear that Peirce abandons true continuity by denying infinite divisibility, and replacing it with infinitesimals . And, it is also clear that this procedure is logically incoherent. In order to have real existence, the infinitesimals require real boundaries. So if the infinitesimals are real, then the continuity is not, due to the existence of the boundaries. If the infinitesimals are produced from arbitrary divisions created by us, then there is no principle by which infinite divisibility is denied.

Aristotle's cosmological argument denies the possibility of prime matter (the fundamental potentiality), as an asserted possibility, which is actually impossible. That is the force of the cosmological argument, it demonstrates that the concept of prime matter is incoherent in relation to empirical evidence.

Peirce's "infinitesimals" may be consistent with "prime matter", and it may be the case that Dons Scotus supported the concept of prime matter. This would place Peirce as closer to Scotus than to Aquinas, who supported Aristotle's cosmological argument. Aquinas supports the logical need to position God as the actual eternal being, denying the eternal potential of prime matter, thus justifying true infinity.

I'm not as familiar with Scotus. I found him rather shallow and uninteresting in comparison to Aquinas, so I focused more on the latter.

Did you read my last post? I quoted from your referenced article on Peirce's synechism, and offered a critique from my own "Thomistic" perspective.

I would think that conditions not to dissimilar to what we have now would be the closest we could come to ok (or are you saying that this is also not ok?). — Punshhh

The climatic conditions change from day to day, season to season, year to year. What would you mean by "not too dissimilar to what we have now"?

Once large, or rapid global changes start to happen ( I'm not saying they will necessarily), we will, I expect, discover that the climactic conditions we have been used to for the last few thousand years were remarkably stable and settled and that they would rapidly become unstable and extreme, relatively. — Punshhh

The last ice age ended about 10,000 years ago, and there was a "Little Ice Age" in medieval times, so I don't know where you get the idea that the climatic conditions have been "remarkably stable" for the last few thousand years.

Again, I am not a Thomist. but I am hoping to find a philosopher here who has studied Thomism and Duns Scotus, and who is willing to delve into the differences with me. — Mapping the Medium

It appears like you want someone to agree with you, not someone to delve into "differences".

For anyone interested, here is the subject to be discussed:

These three varieties of evolution Peirce renames, respectively, as: tychasm, anancasm, and agapasm (using the related Greek roots to provide a technical terminology). The first two, he claims, are degenerate forms of the agapastic: that is, while each is a real evolutionary force, the reality of the evolutionary universe as a whole is comprised by the third form. While tychasm finds growth from the lower into the higher a matter of luck (as well as “lower” and “higher” being purely circumstantial adjectives), and anancasm sees it as a matter of internally-driven necessity (and is thus a Whiggish theory of nature, at heart), agapasm sees it as “a love which embraces hatred as an imperfect stage of it”; which seeks elevation of the lesser through a not-yet-realized better. That is: “Love, recognising germs of loveliness in the hateful, gradually warms it into life, and makes it lovely. That is the sort of evolution which every careful student of my essay ‘The Law of Mind’ must see that synechism calls for.”
This, as Peirce calls it, is creative love. It is not a love which seeks fulfillment of itself, but which calls out for as-yet-unrealized perfection. It is love as a final cause: first in intention, last in execution, the cause that makes anything to be at all. It is the cause that answers the question “why?” for anything.
Few people already convinced that evolution proceeds through random chance will be persuaded of its inherent purposiveness, let alone that this purposiveness is not itself the product of chance — it echoes too loudly of a theistic hand guiding the universe; and natural purposiveness implies all sorts of normative consequences, including moral ones.
The challenge that Peirce’s synechism issues us, however, is this: if the universe really is found to be continuous, such that between any two things there is no unbridged gap but a gradient of infinitesimal degrees of difference — in at least potency if not actuality — if this continuity exists in fact and not only in theory (and a careful examination, I think, can only lead one to the former conclusion): what then explains this continuity, if not agapasm?

The problem with Peirce's metaphysics is that he allows that pure, absolute continuity, which can only be expressed by us human beings through the terms of infinity, to be polluted by the concept of "infinitesimal". A succession of infinitesimal points does not provide the necessary conditions to fulfil the criteria of "continuity". Positing a degree of difference as existing between the infinitesimal points, no matter how large or small that degree of difference is, necessitates the conclusion that there is something "change", which occurs between such points, rendering the supposed continuity as non-continuous. To assert that such a difference is a difference which does not make a difference is to assert contradiction.

Because Peirce proposes a polluted, and impure form of continuity, rather than starting with a pure and true continuity as his first principle, his approach to agapasm is demonstrably a materialist approach. And, he provides no bridge between his materialist foundation he provides, and the true spiritual "Love" which he espouses, because his synechism is a false, deceptive synechism.

I've been told that if I truly understood calculus, I would see how there is no contradiction in something spatially being finite and infinite at the same time. I suck at math so I could be the stupid one in the conversation :( — Gregory

In mathematics, to "truly understand" is to accept the axioms without question. This allows that contradiction within an axiom is acceptable as understood. Blatant contradiction is not the real problem though, rather ambiguity and vagueness, such as the difference between "continuum" and "continuity", the definitions of "object" and "infinite" are the real problem.

Philosophers have an unfortunate tendency to mistake ordinary uses of equality as denoting a physical relation between things rather than as being a linguistic relation between terms. — sime

This is why "x=x" is not a good way to express the law of identity. It really doesn't serve that purpose.

Nice try at saying everything will be ok after all. — Punshhh

That's not what I was trying to say. I was saying something more along the lines of "for many of us there is no such thing as ok".

I think that "infinity contained within the finite" is contradiction pure and simple. That's like saying that a finite thing is inherently infinite. Contradiction.

I think one of the problems is that when you hear that the earth’s temperature might warm by 2.5 degrees, a lot of people say ‘so what? Temperatures change by more than that every hour.’ They don’t realise the fundamental importance of what used to be called ‘the balance of nature’. — Wayfarer

Also such developments could affect the temperature conditions of the earths crust resulting in seismic and volcanic activity. — Punshhh

Volcanic activity can put a huge amount of ash into the atmosphere, lowering the atmosperic temperature significantly. Maybe the "balance" has not yet been lost and we've yet to see the swing back the other way.

I did suggest though that climate change may be that replacement. It offers a relationship with something bigger than ourselves, which is the planet, life and the universe. Climate change is a quasi religion that promises a better world, a closer, more meaningful relationship with the environment. It’s message and quest are beyond question; who would not think it imperative to save the world, who would not want to embrace such a beautiful existence?The future, once we overcome climate damage, is golden, Edenic, perfect in its balance between needs and resources, everyone happy, everyone taking only what they need, everyone giving and sharing. An end to capitalism, an end to greed, an end to poverty. — Brett

The human being, "man" in particular, has always had a fascination with the idea of exercising control over "nature", complete dominance. It's sort of a fantasy, which with the aid of science, has developed into an illusion, that we actually can have control over the natural world. In the past, the illusion has always been shattered when "the hand of God" strikes, Noah's flood for example. Replacing "God" with the more submissive "mother nature", is the first step from fantasy to illusion, or even delusion. The concept of "climate change" is another such step. We produce the science which shows that we have affected the climate to the extent of X degree, which validates the claim that we as humans can change the climate. Then we argue that such and such actions are needed to negate this affect, bringing the climate to a "normal" state, thereby propagating the illusion that we might exercise control over mother nature. This is well received by those with the urge to dominate.

Now, the problem is that since this is rightly described as an illusion, or even delusion, the people who see through this illusion have no desire to act on this premise, regardless of how dangerous the pollutants actually are. So the premise of this "climate change" movement is faulty, because it cannot get action from the people it needs to get action from. If instead, we address the various pollutants such as CO2, and describe exactly why the pollutant is harmful, and why emissions ought to be controlled, rather than launching into nebulous ideas about human beings having the power to change the climate, the movement would probably have more credibility.

A length that is irrational comes into play when you have a length that is the "smallest" length as the right sides of the triangle. The irrationals are not imaginary numbers. They simply go on forever, within a limit. — Gregory

The irrationals are relations which cannot be resolved, like the ratio between the circumference and the diameter of a circle, or the distance between two points of equal distance from a point at a right angle. That such relations are irrational indicate that the two things being related to one another are incommensurable.

Your math is wrong. — fishfry

Yes, sorry, I misunderstood.

The irrationals fill in all the holes in the rationals. I already illustrated this with a sequence of rationals that approaches the point sqrt(2) but there's a hole there instead of a point. The irrationals fill in those holes. — fishfry

There are no holes in the rational numbers, just like there are no holes in the reals. The numbers are produced from a different set of rules. This is what your illustration illustrates, it does not illustrate a hole:

But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry

What you illustrate here is that an irrational number cannot be expressed in the rational system. If we did not already know what the sqrt(2) is, as an irrational ratio, we would not know that your proposed rational sequence does not converge to that. You could make up any random, fictional, irrational number (not grounded in a true irrational ratio like sqrt(2) or pi), and show that a sequence of rationals does not converge. But this does not illustrate a "hole". it just illustrates some sort of incompleteness, like 1.5 (3/2) illustrates an incompleteness in the reals.

The reals fail to account for division. The rationals are an attempt to address this failing, but to the extent that there are irrational ratios, the rationals fail in the attempt to account for division.

So you could make another demonstration to show that there are rational numbers which cannot be expressed in the real system, but this does not indicate "holes" in the system. Such demonstrations, which demonstrate incompleteness do not illustrate "holes", they just show that the different systems follow different rules. There is not one set of rules to cover all mathematical operations. If there is incompatibility in the rules, it will appear as what you call a "hole".

The existence of irrational numbers is easily accounted for by the fact that in the rational system (rules), there is infinite possibility of numbers between any two rationals. "Infinite possibilities" is not within the bounds of any logical system, it refers to where the system fails. Therefore illogical divisions (irrational ratios) are allowed to exist within that system, which allows for infinite divisibility. The illogical divisions are within the rules of the rational numbers, system. So the irrationals do not represent holes, as if they are outside the rational numbers system, they are within the system, a manifestation of the illogical proposition of "infinite divisibility". "Infinite possibility" is itself illogical. As a rule, "infinite divisibility" allows for the illogical, i.e., for the irrationals to exist within the rational system.

I could drill the math down a lot more but should probably wait for encouragement, and if none is forthcoming I should leave it be. I don't think you're curious about the math at all. You just want to throw rocks. But why? People uninterested in chess don't spend their lives hating on chess. They just ignore it. You think math is bullshit? Maybe you're right. Maybe it is all bullshit. The thing is why do you keep repeating the point over and over as if we haven't all heard you already? And as if we all don't already understand the point? — fishfry

That's right, I'm not curious about that math because I think it's the wrong approach. approaching the irrational numbers as if they are holes in the rational numbers, and trying to fill those holes with the reals, is completely backward. If these are actually "holes", then the reals are like a sieve. There's a massive "hole" whenever an odd number is divided by two for example.. Why would you try to patch holes with a sieve?

To resolve a problem requires a clear, and complete analysis of the problem itself, to understand its true nature. This is what I offer, a more thorough analysis of the problem, as an aid, to assist in resolution of the problem. What I do is not a case of throwing rocks, or hurling insults, it's a case of examining the foundations for weak points. If your attitude is that these foundations were built by the greats, therefore there are no weak points, (appeal to authority resolves fundamental problems), then I think you are in need of God's help. Hopefully, God as the ultimate authority, will show you how his principles differ from those authorities which you appeal to.

Care to describe this so-called "third window"?

Fully understanding nominalism and ontological individualism, where we took a drastic, misguided turn in the Middle Ages, and then revisiting Duns Scotus, is crucial to understanding much of Charles S. Peirce's thought. ... — Mapping the Medium

I don't see why you say that this is a drastic misguided turn. Further, I don't think Peirce represents this as a drastic misguided turn, so I don't see the association you are making here.

America was founded during a time in history when individual rights were front and center, and Descartes' "I think therefore I am", and mind/body dualism, was encouraging a freedom of individual thought, separating and elevating humans to a realm seemingly above nature, theologically in an attempt to understand the mind of God. We lost sight of the importance of shared understanding. Everyone wants to be right, when no one is. The 'Medium' is always cloaked, unless we interact with each other through dialogue toward a shared understanding. This has all caused us to get further and further apart, encouraging divisiness, hatred, etc.. We are now dealing with screen infested, narcissistic demands, and less and less cooperation and dialogue. ..... I hope this explanation helps a little. This is 'ontological individualism'. — Mapping the Medium

That an individual is in a very certain, and real sense, isolated from others, is a fundamental brute fact. To deny this isolation, and emphasize the reality of a united humanity, society, or some such thing, is to belie the true essence of the "Medium".

There are two approaches to this "Medium". One takes the Medium as a natural separation between individuals, and the other takes the Medium as a natural unification of the whole, humanity, or society. The latter is clearly wrong. The Medium exists as a natural separation between the individuals, and it must be manipulated artificially, through language, construction, manufacturing, production, and other institutions like those of education, to create the unity which we call society. That the Medium might become a force of unification, when its essence is separation, is the result of the inspired efforts of ambitious free willing individuals. To deny this, and claim that the Medium is a natural force of unification is to deny the reality of free will.

The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not. — aletheist

This is exactly why it is a mistake for us to try to discuss "real space", and "real time". If physicists produce a concept of space and time as a continuum, in the mathematical sense, yet we want to say that "real" space and time are continuous, and this is different from the mathematical continuum, to talk of "real" space and time is a mistaken approach. It is mistaken because there is no such thing as real space and time, these concepts are derived from the observations of objects which are assumed to be real. So the correct approach would be to say that physicists incorrectly model the existence and movement of objects, and that they ought to be modeled as continuous rather than as a mathematical continuum.

Notice, that what is being modeled is the movement of objects, and these are what are assumed to have real existence, not space and time. "Space" and "time" are produced from these models, as logical conclusions. Model A shows objects to move in such and such a way, therefore there must be such and such "real space and time", to substantiate that model. But model B shows objects to move in a slightly different way, so there must be a slightly different "real space and time" to substantiate that model. The "real space and time" is just something created by the model, as an incidental conclusion. This is why model-dependent realism is popular. But if the model is the "correct" one, then there is a real space and time of such and such nature to support that model. If there is no "correct" model, as model-dependent realism proposes, there is no real space and time. Space and time are not themselves being modeled. You can see this in the difference between Newtonian principles and Einsteinian principles. Movement of objects is modeled, and a "real" space and time of such and such a nature is required if the model is supposed to be true.

Ultimately, we might produce a model of moving objects which required no space or time, but our present conceptual structure falls back onto the reliance of space and time. However, the fact that our conceptual structures of the movement of objects requires that space and time are real, does not mean that space and time actually are real. If you and I agree that there is misconception, related to the use of "continuum", and "continuous", then the conceptual structures are inaccurate, and therefore a conception without any real space or time might be the correct one.

This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves. — aletheist

There's no such thing as real space and times. We see things, we create things, distinct objects, and we see how things move around and change, and we make these concepts of space and time to account for these perceptions. What would real space, or real time even be like? We have no such concept, of real space or time.

What the concepts of space and time represent, is our understanding of the existence and movement of objects. There are not any things represented by "space" and "time". Kant covered this thoroughly, these are fundamental "intuitions" he called them, which provide for us the possibility of understanding the physical existence of things.

That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. — aletheist

This is the commonly accepted definition of "continuum" in mathematics. You can't say that the mathematical definition is wrong, because it's a mathematical term. It's not used anywhere else, so when mathematicians use the term, this is what they are talking about. What we have to respect is that "continuum" is not the same thing as a "continuity". If you think that "continuum" ought to be used to refer to a continuity, then it's you who is wrong, because this is not how the word is used in mathematics, which is its normal place of usage.

Only in the sense that continuity is a property, while a continuum is anything possessing that property. — aletheist

This is not true, and that's the point I've been trying to make. A "continuum" consists of a series of distinct elements, whereas "continuity", or "continuous" refers to one uninterrupted whole. Therefore a "continuum" as the word is commonly used in mathematics cannot be continuous, nor can it have the property of continuity. If it had the property of continuity it would be one continuous thing, and not a series of distinct things, as "continuum" implies.

What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself. — aletheist

Space and time are concepts derived from our observations. Therefore it makes no sense to say that either space or time extends beyond what is observable. What we call "space" and what we call "time", are abstract ideas created to account for what we observe. If some aspects of reality are beyond what is observable, then they are neither spatial nor temporal because these are empirically based concepts.

I fully agree that there are aspects of reality which are not observable, but these are non-spatial, and non-temporal aspects of reality, as "space and "time" apply to observations.. That there are such non-observable aspects of reality, is what validates the concept of the "immaterial", and "non-physical".

By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. — aletheist

The problem here is the difference between a continuum, and a continuity, which I explained already. "Continuum" as implied by common usage means a collection of contiguous, but separate individual units. "Continuity" implies one uninterrupted whole. So if we assume "a whole", it could be composed of separate parts, like a continuum, or it could be a continuous whole, a continuity, but it cannot be both, because of contradiction. So to speak of a whole, as a continuity, is to speak of one thing, and to speak of a collection of parts, as a continuum, is to speak of a completely different thing. "Continuum" and "continuity" have very different meanings.