So, you separate the intellect from the senses by virtue of positing a mind(presumably of God) and then tell me that my claim that senses precede intellect needs justification? — creativesoul

I have not presumed God, I just gave you the logic. A sensing body is an organized body. This means that it requires ordered parts. The only thing which is capable of ordering parts, is an intellect. Therefore intellect must precede sense. Thus my claim is justified. Yours, that senses precede intellect, has not been justified.

From the Platonic perspective, which is what I am giving you, the immaterial mind as "soul", precedes the material body and causes the parts which constitute the organized body to be ordered in the necessary way. There is no need to assume God at this point, only the need to assume an immaterial soul, as prior to the material body. This is because a material body can only exist as an organized body, and that requires something to order the parts.

It is only when we consider the belief that material bodies preexisted life forms, that we see a need to assume God. This is because these material existents also exist only as ordered parts, and some sort of intellect is required as that which orders the parts.

Which tool do we use without requiring us to trust and use our senses? Which thought can we have without using our senses? — creativesoul

I don't see how these questions are relevant. Questions do not justify your claim, nor do they address the logic I've presented you with.

I just took a moment doing what I do to read this post, and now I feel so guilty. :cry:

You have no mercy, MU. — jgill

That's right. Show up at confession and feel the guilt for your sins. But you know you will be forgiven.

They're comparable because in each case the number remains the same. On that basis you reject that a negation has occurred but, apparently, still accept that an addition and a division has occurred. Which seems to be an arbitrary conclusion. — Andrew M

There is nothing about the definitions of "addition", or "subtraction" which requires that the result be other than the starting number. "Negation" is defined as producing a statement other than the one which is negated.

I've linked to the definition several times now. Here it is again with the relevant parts bolded. — Andrew M

You provided a definition of "additive inverse", not of "opposite", nor of "negation". And, as I've told you already, your quote only demonstrates that mathematics uses these terms in a way which is inconsistent with other fields of study, like philosophy and logic.

In fact, I see now that there is inconsistency within the quoted paragraph itself. It says: "For a real number, it reverses its sign". And it also says: "Zero is the additive inverse of itself." Since zero is a real number then it is an exception to the stated rule for real numbers, therefore the inconsistency inheres within your definition. It is self-contradicting, stating a rule then a contradicting rule.

For example, to negate 2 is to subtract 2 from 0 which is -2. Conversely, to negate -2 is to subtract -2 from 0 which is 2.

Similarly, to negate 0 is to subtract 0 from 0 which is 0. — Andrew M

This clearly demonstrates the contradiction. Negating a real number is to reverse its sign, by your definition. Zero is a real number. Yet you propose that zero is negated without reversing its sign.

You seem to be forgetting what negation is:

In logic, negation, also called the logical complement, is an operation that takes a proposition {\displaystyle P}P to another proposition "not {\displaystyle P}P", written {\displaystyle \neg P}\neg P, {\displaystyle {\mathord {\sim }}P}{\displaystyle {\mathord {\sim }}P} or {\displaystyle {\overline {P}}}\overline{P}. — Wikipedia

Notice, negation takes the proposition to "another proposition". There is no exception, which would allow that a negated proposition could remain the same, as you propose with zero. You simply refuse to adhere to the rule, and insist on defending all those sinners who have gone before you. Please, approach the confessional box, now! You will be forgiven.

Welcome to the world according to Banno.

Do you also hold that adding zero to a number cannot be called "addition" because the number is the same before and after?

Or that dividing a number by one cannot be called "division" because the number is the same before and after? — Andrew M

Why bring up things which are not comparable?

Yet "negation", defined as zero minus a number, can be just that. — Andrew M

Well, I've never seen "negation" defined as " zero minus a number". Care to share where you got that one from? Zero minus a number clearly does not negate the number, as negating a number gives zero. So I think you are really stretching for straws now Andrew.

Where do you get this from? This is not how mappings work. — Real Gone Cat

We are not talking about maps, we are talking about reflections, mirrors. It's your analogy, keep on track and don't change the subject please.

Here, we're considering a single plane of reflection, and a single reflection (a single mapping). You've invented a situation that doesn't exist. — Real Gone Cat

No, you are changing the analogy. There is no "mapping" in the analogy.

I imagine you're a wonderful person, so it pains me to have to say this : usually, discussing math with you is like discussing the phases of the moon with a flat-earther. You really have no idea what mapping, or inverse, or almost any other math term means. And you have no interest in learning. — Real Gone Cat

Again, you are trying to change the subject. We were not talking about "mapping".

What's truly odd is that you're lack of understanding is at the most basic level. You stumble on understanding simple facts about the integers and zero. The Chinese and the Hindus understood the nature of zero thousands of years ago, and even late-to-the-game Europe has known about zero at least since Fibonacci's Liber Abaci. No one debates this stuff anymore. — Real Gone Cat

Please don't be an asshole Real Gone Cat. I really don't understand why some people get so upset when the axioms of mathematics are debated. There's no reason for it, it's just a field of study. Keep your shit together

In the discussion we've been having, the integers (positive, negative, and zero) are clearly a group under addition, with the identity element being 0. So by the theorem above, 0 is its own inverse. — Real Gone Cat

What I am arguing is that mathematicians ought not accept such theorems, I am not trying to say that they don't accept them. So, you providing me evidence that they do accept them, just provides me with inspiration to produce a stronger argument that they ought not do what they do.

It is neither. The negation of zero (a number without a sign) is zero (a number without a sign). The number does not change. — Andrew M

The point is that this cannot be called a "negation". If the thing, zero, is the very same prior to, and after, the proposed "negation" then there has been no negation. "Negation", by definition, creates a statement which is distinct from that which is negated. There cannot be a "negation" with the negation being the very same statement as that which was negated. This cannot be called a "negation".

You need to be more specific. What flaws and differences? — Andrew M

Uh... we're discussing one right here, for example, the role of zero.

Plato's principal message amounts to setting an unattainable criterion. The intellect follows from the senses. The senses are primary. The intellect is secondary. — creativesoul

For Plato it is not an unattainable criterion, it is a description of reality, what is the case. The soul necessarily precedes the body as the cause of order in the material parts which is what constitutes a living body, organized parts. So in Plato the mind is prior to the body, and must rule over it to maintain the order of the parts. That's a fundamental tenet of Plato's dualism, repeated many times. And he posits a third thing, passion or spirit, as intermediate between body and mind, and the means by which the mind rules the body. This third thing accounts for the supposed "problem of interaction" commonly charged against dualism. If the fundamental order gets reversed, and passion or spirit is allowed to ally itself with the body instead of the mind, and the intellect is allowed to follow the senses the result is irrational acts.

Your claim, that the senses are primary, and intellect follows from the senses needs to be supported, justified. The problem is that sensation requires ordered material parts. And nothing but a mind or intellect is known to be capable of ordering parts.

This is entirely your own invention. Give one citation to support this. Just one. — Real Gone Cat

There's a lot written there. Let me know what you think needs to be supported, and I'll address it. Do you not believe that a reflection is light rebounding from a surface? Or what exactly is it that needs to be supported? Do you not believe that if a spot on the reflecting surface reflected back on itself, that this would create an endless back and forth of the light reflecting between the spot and itself, analogous to two mirrors facing each other?

This is an example of you digging in your heels. You're so math-phobic you have to invent concepts out of the blue to justify your stance. But "you know what you know". — Real Gone Cat

Yes, digging in my heels to stand up for what is real, rather than to fall for some smoke and mirrors deceptive proposal from someone like you. Whether you call your proposal math, physics, or some other type of science, I will stand up against it when it is obviously untrue.

Point out where I said zero is both positive and negative. Here, let me help you : — Real Gone Cat

It's implied by the very position you are arguing. The negative numbers are said to be inverse of the positive. And by your analogy, the negative "reflect" onto the positive. So if zero "reflects" onto itself, it must be both negative and positive. There is no other possibility when we are talking about the negatives reflecting, or being "across from" the positive, if zero "reflects" onto itself, then it must be both. If zero is across from itself, like the negatives are across from the positives, then you are describing it as being both negative and positive.

It's understandable that you might not be inclined to say that a person who has no apples has a certain number of apples, namely 0. What you'd prefer is to say that they do not have any apples. There is no quantity that they have at all, and calling 0 a quantity is an abuse of the idea of quantity. That's understandable. The same with measurement: to say that a person who takes one step to the right has moved that amount is fine, but it is an abuse of the idea of distance to say that a person who has not taken a step at all has moved 0 steps to the right, to the left, whatever direction you like. — Srap Tasmaner

You are not quite representing the complete extent of the issue here. The problem is not in calling zero a quantity. That is an acceptable move. The problem is in accepting the consequences of this move, what it implies about the nature of "quantity" when you allow zero to be a quantity.

When zero is a quantity then it falls into the same category as the negative numbers and the positive numbers. Each number signifies a quantity and so does zero. Then we cannot express the numbers as having a mirror opposite, or inverse, the negative numbers being inverse to the positive, because there is a number, a quantity, which has no inverse, zero.

So if the desire is to represent the negative numbers as an inversion of the positive, then we must represent 0 as distinct from the numbers, just like the reflecting surface, or mirror, is something different from the arrangement of light. This points to the difference between cardinal numbers and ordinal numbers. If, what is expressed by a number is a position in an order, rather than a quantity, then zero can be apprehended as a complete lack of order, and this distinguishes it from the numbers which represent order, but then it cannot have a position with the other numbers, on the number line.

It's the bag, the difference between not having a bag at all and having a bag with nothing in it. 0 ends up playing a prominent role in positional number systems because the positions in such a number system are like bags laid out on a table into which you can put at most a certain number of items. But the bags are fixed; you do not remove them when they are empty.

Similarly, when we do algebra, we use containers for values, variables, and it may be possible for a variable to hold no value at all, that is, 0. But the mathematical functions we apply to a variable are defined so that they go through even if turns out the variable held a value of 0, or no quantity at all. You just have to follow some rules, so that you don't mistakenly divide by 0, which makes neither mathematical nor intuitive sense, as in this famous 'proof' that 1 = 2: — Srap Tasmaner

Your use of "value" here seems ambiguous. A value could be a quantity, or it could be a position in an order (hierarchy, or priority). You have not clarified which of these "the bag" in your example, represents. At first you talk about "position", such that the empty bag has a position. But then you say that a value of zero has no "quantity". To me, the latter makes sense, but not the former. It makes sense to say that the empty bag is a container with the quantity of zero apples. The empty container represents that quantitative value.

But the case of position is not so straight forward. If a position is represented by what's within the bag, then the bag itself is not representative of anything, and all bags are the same, as irrelevant. So the empty bag represents a position, through its emptiness, and that must be no position whatsoever (no order). This implies that the empty bag, zero, or no position relative to the order, has no place on the number line, or, is equally well positioned relative to any place. And that it has no place is well represented in practise by the fact that we can count forward or backward starting from any number, we do not need to start at zero. The counterintuitive thing though, is that we should not ever hit zero in counting like this. So if we count down from 2, it should be 1 next, and -1 after that, skipping the habitual "0" here, because zero has no place in the order.

This all relates to how we apprehend point zero, or t-minus zero, in the temporal sense. If we relinquish the idea that there is an exact, zero point, we can remove zero from the number line all together. Then 1 and -1 are directly opposed to each other, and the presence of the two mark the division between positive and negative, as the first position on each side.

Okay, I'll give it a go. But you usually dig your heels in and refuse to hear otherwise when it comes to math. Try to have an open mind.

I could offer an intro to group theory to prove zero is an inverse of itself, but I don't think that's going to sway someone so math-phobic. Let's stick with the idea in my previous post : Can we agree that "opposite" sometimes means "across from"?

To be across from something means to be reflected in a line, point, or plane. Even when facing a friend at a table we can be said to be reflected in an invisible plane between us (actually reflected in a line to preserve left- and right-handedness).

What's of interest is what happens to points lying on the line (or point or plane) of reflection. Under the reflection, such points do not move! Thus a point on the surface of a mirror will reflect onto itself!

When a reflection in zero is performed on a number line, every point maps to it's negated version, but zero maps to itself. In other words, zero is across from (opposite to) itself. — Real Gone Cat

The analogy really does not work Real Gone Cat. A reflection is light rebounding back off a reflective surface, which you represent as a plane. If there was a spot on the reflective surface, a point on the plane, which reflected back on itself, "itself" being a point on the reflective surface, would reflect it again. back on itself, and again and again. This would create an endless back and forth between the spot and itself. This is like having two mirrors in front of each other, accept that your proposal builds this right into the single plane, or reflective surface..

If such spots existed on the surface, each spot would effectively annihilate the capacity of the mirror to properly reflect at that point because the reflection would get absorbed into the infinite back and forth with itself. So if the rules of mathematics allow that zero "maps to itself" in this way, this would effectively annihilate the integrity of the concept "zero", as such a reflective surface (separation) between positive and negative, just like a spot on the mirror reflecting back and forth on itself would absorb the light and not reflect outward, ruining the integrity of the mirror as a reflective surface.

I think this is what is alluding to. If we allow that "zero" implies both positive and negative (in a self reflecting way) in common applications, instead of neither (as we actually do), this would destroy the integrity of "zero"

They are different kinds of inversion. What would a "true inversion" be? — Andrew M

This is exactly the point, there is no such thing as a true inversion. Inversions are not real things, just like symmetries, they are ideals. So in the realm of the ideal, like mathematics, we can stipulate, or propose something like "an inversion", or "a symmetry" and we can convince ourselves that such proposals or stipulations provide a real, or true representation. But when we get down to the nitty gritty, of analyzing the representation for accuracy, we see the flaws, the differences between the supposed representation and the thing represented.

When we see the existence of such flaws in the representation, we ought to acknowledge that the ideal, the proposal or stipulation, is not meant to be a representation at all. The ideal, in this case "the inversion", or "the symmetry", is not meant to represent reality in any way, it is a tool which we apply toward reality, in a sort of comparison. We can then see where reality varies from the ideal

We can learn from this process of comparing the proposed ideal to reality, but it is necessary to determine where there are flaws in the proposed or stipulated ideal, i.e. where the proposed ideal is less than ideal. This is necessary because we need to know whether it is the case that the differences between the proposed ideal, and the reality, are due to the reality being less than the ideal, or the proposed ideal being less than ideal. When reality appears to be different from the ideal, we tend to think that this is because reality is less than ideal. But if there are deficiencies in the proposed ideal, it could be the case that reality is more ideal than the ideal, because the ideal is really less than ideal.

So in this case, we can see that the proposed ideal, is really less than ideal, because the proposed inversion is contaminated by the presence of zero on the number line. As would say, the zero is a piece of poop in your mathematical sandbox in this proposed "inversion". Allowing zero on to your number line makes your inversion between positive and negative numbers less than ideal.

Major Edit : "Opposite" is perfectly fine when discussing positives and negatives. One of the meanings of opposite is "across from". Consider the number line with zero as the value between the positives and negatives. +5 is across from -5. Opposite works. — Real Gone Cat

That doesn't resolve the problem, which relates to zero being a number, as having a place on the line. Is zero across from zero?

I suggest that these false inversions, which are inversion-ish, rather than true inversions, are what create the appearance of symmetries in the application of the mathematical principles which describe something which is not a true inversion as an "inversion". Then symmetries are taken by some philosophers, to be something real, existing in the universe, instead of just a product of the mathematics, and misleading descriptive terms. This has opened a whole new field of speculation into an assumed phenomenon known as "symmetry breaking". But the symmetries are just fictional, imaginary, produced from the misuse of descriptive terms, and so that speculative field of symmetry breaking speculates about the activity of things "symmetries" which don't even exist.

But I knew it was true when I wrote that post a few hours ago, so it's not only now that I know it to be true. — Banno

Isn't the now of a few hours ago the same now as the present time? Or do you have the capacity to divide time into a multitude of distinct, particular, and separate nows, such that a past now would be distinct from the present now?

In math we also have inverses, additive and multiplicative. They're opposite-ish, the way equivalence is equal-ish. — Srap Tasmaner

This was the point I was making in the first place. Additive inverse is different from multiplicative inverse, because neither represents a true inversion, they're inverse-ish, each in its own specific way.

t's a pretty standard thought, at least in eastern philosophies, that the self is an illusion. — T Clark

Then how would you even begin to talk about sensations like hearing, seeing, etc., if there is not something doing the sensing? If you have an aversion to the term "self", that's one thing, but isn't it still necessary to assume something which is sensing, in order to make sense of sensation?

I don't think the sensations are "what are real", i.e. all that is real. I think they are the measure, or at least one measure, of what is real.

If we start from human sensations, shouldn't that which is sensing be just as real as the thing sensed?
— Metaphysician Undercover

Are you asking if we, our selves, are real? It's a good question. I didn't address that in my OP, but I didn't intend to exclude it from the discussion. — T Clark

Right, that's the point. We consider whether or not the thing being measured (through sensation) is real, and we naturally conclude that if we are measuring it, it must be real. But prior to coming to this conclusion, isn't it necessary to do our due diligence toward understanding the thing which is doing the measuring? If the thing doing the measuring isn't real, then what validity does "if we are measuring it, it must be real" have?

No, I am saying that particular collections are made up of particular collections, not constructed from universals. I take particular collections as granted because I see them all around me and because for any particulars there necesarily seems to be a collection of them, and universals don't seem necessary to explain the existence of particulars. — litewave

That's an irrational, infinite regress, which we already discussed when you said that an object is a collection of objects. The problem is that you've created a vicious circle by saying that a collection is made of collections, and you have no indication of what a particular is. A "collection" is a universal, a group of many. Now you want to deny that a collection is a universal, and claim that is a particular.

You claim to see collections existing as particulars all around you. Please explain to me how you think that you are seeing a collection as a particular when you haven't even said what a particular is. Perhaps an example or two?

For most people, for most concepts, acquaintance with instances of the concept precede, in time, the possession of the concept, and exposure to those particulars is instrumental in acquiring the universal they fall under. That's the argument from ontogeny: you are acquainted with moving, barking, licking particulars before you know that they are dogs. And there is a related argument from phylogeny: modern humans have a great many concepts that they were taught, often through the use of exemplars, but it stands to reason that not every human being was taught: there must have been at least one person who passed from not having to having a concept unaided. In essence, we imagine that person somehow teaching themselves a concept through the use of exemplars, and we imagine that process proceeding as we do when analyzing a population of objects, looking for commonalities. — Srap Tasmaner

Yes, this is the effect of teaching, learning. From the perspective of learning, we see the particular as essential to learning the universal, because this is the process which taught us. However, the particular is a tool of the teacher, who already understands the universal to be taught. So from the learner's perspective, the particular appears to be essential to the learning process, as necessary for it, but it is really a weak sense of "necessary", as what has been determined by the teacher as needed, required for the process. It is not a true logical necessity because it might be possible that the student could learn the universal in another way.

This is what Plato looked at in The Meno, with what is referred to as the theory of recollection. The student is induced to produce the universal without the use of a demonstration with particulars, and the observers conclude that the student must have already somehow had the universal in his mind. So they propose, as a solution, that the student must have somehow had the universal in his mind, from a past life, and recollected it. You can see that the proposed solution is inadequate, but it gives us a good representation of the problem. Aristotle gave a better solution, by saying that the student has within the mind, the potential for the universal, prior to actually formulating it.

But use of the particular, as a teaching tool necessitates in a stronger way, that the existence of the universal to be taught preexists the use of the particular through the concept of causation. And if the potential for the universal, which precedes the actual existence of it in the mind, does not necessarily require particulars for its actualization, then what does constitute the actual existence of the universal?

hat's of interest here is that resemblance is not only relative, but comparative: resemblance is a three-way relation, a given object resembles another more, or less, than it resembles a third. — Srap Tasmaner

I agree, there always must be a third in this form of comparison, because two will always be other than each other. But this only demonstrates that "resemblance" is not the true principle by which we categorize. In reality, we produce the category, like "dog" in your example, and judge the thing directly as to whether it fits the category, without comparing it to others within our minds So you see an animal and call it a dog, without performing mental comparisons. And learning the category is a matter of developing the capacity to do this, not a matter of learning how to compare. That's why learning the category is the important aspect, and it consists of seeing examples, not of comparing three things.

The material of the Sesame Street skit is only used to demonstrate that the category has been learned. That's why it gets sort of controversial, because to demonstrate that one knows the group, a person is asked to say what is not part of the group, as a simple form of confirmation. In reality an act of exclusion is not necessary if one has learned the category. We simply need to judge and include members as a part of the group without indicating what is not a member. This is like determining what pleases you without any reference to what displeases. And the commonly touted principle, that one must know "what X is not", in order to know "what X is" is a false principle. It seems to be based in the faulty idea that one must demonstrate one's knowledge, to have it.

I have, twice. But here it is again with the relevant parts bolded:

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change,[2] and negation.[3] For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
— Additive inverse - Wikipedia — Andrew M

That's not a definition of opposite, just a use of "opposite" which clearly demonstrates what I said. Your definition of "inverse", as "additive inverse" renders the meaning of "opposite" as inconsistent with common usage. You've demonstrated that by using this definition of inverse, zero is opposite to itself. But a thing being opposite to itself is contrary to common usage of "opposite".

You should try that. What happens if you have a -0 unequal to 0? — Srap Tasmaner

A better thing to try is to consider what happens if a thing is allowed to be opposite to itself. Opposites are commonly the two defining extremes of a measurement scale, hot and cold, big and small, etc.. If we stipulate that the two extremes are the very same thing (like zero relative to the scale), then we have no way to distinguish whether some thing which we're trying to measure, but is off the end of our scale, is off the top of the scale, or off the bottom of the scale, because we have set the conditions whereby the two are the very same (zero relative to the scale).

I really don't think you've provided any "mathematical definition of opposite". But if the mathematical definition of "opposite" allows that a thing is the opposite of itself (as zero is the opposite of zero), then yes, I would say that my preferred definition of "opposite" (the common use of the term) precludes the mathematical definition.

Even though the subject we are discussing is mathematics. — Andrew M

I suppose you ought to produce this mathematical definition of "opposite" so that we can judge whether zero is truly opposite to itself, by that definition. Or whether it is really the case that your preferred definition of "inverse" renders the common definition of "opposite" as inconsistent.

Here's a brief demonstration to help you understand what I am saying. Assume the smallest possible positive number is directly opposed, or inverse, to the largest possible negative number. In other words, we get as close to zero as possible on both sides, and maintain a balance of opposition between the two sides.

Now, let's assume that the quantity represented on each side is so near to nothing (zero) that we might be inclined to round it off. If we do such a thing, then the two quantities on each side become equal to each other, and the same as each other, as zero, instead of opposed to or inverse of one another.

Clearly, two inversely opposed and balancing quantities is not the same thing as one quantity, because that would mean that the positive number closest to zero is exactly the same as the negative number closest to zero, rather than having the two opposed to each other.

zero is its own opposite — Andrew M

Sorry Andrew, but "opposites" don't work that way. A thing is the same as itself, it cannot be opposite to itself. "Opposite" requires two.

As I said, that's a fictional, imaginary, representation of what an object is. And if you look back at where I first engaged you in this thread, you'll see that my principal objection to your proposal is that you take the existence of particulars for granted. Then you claim that people construct universals from these particulars which are taken for granted. So the problem here, is that what you have taken for granted is a fiction, and this undermines your entire proposal as completely unsound.

In reality, you have shown that you construct a representation of a particular, an object, from some preconceived universals, set theory, but then you've tried to claim that universals are derived from particulars. However, you have just demonstrated the opposite of what you claim. The notion of "an object" or a particular, is actually derived from preconceived universals, so the conception of universals is prior to the apprehension of particulars.

Reality only makes sense in comparison to what humans see, hear, feel, taste, and smell in their homes, at work, hunting Mastodons, playing jai alai, or sitting on their butts drinking wine and writing about reality. Example - an apple is real — T Clark

If reality only makes sense in relation to human sensations, then why wouldn't you be concerned with the sensations themselves, hearing, feeling, tasting, and smelling? If the sensations are what are real, then we have two conditions, that which is sensing, and that which is sensed. Why do you proceed only toward that which is sensed, the apple? If we start from human sensations, shouldn't that which is sensing be just as real as the thing sensed?

In set theory, ordered sets/collections (which have members arranged in a particular order) can be defined out of unordered sets. — litewave

But mathematics doesn't give us a true representation of what an object is. Math is composed of axioms which are produced from the imagination. That's what I told you earlier, why the relation between two things, described by a universal, need not be a "resemblance" relation, if universals are constructed by the mind. The relation might be completely arbitrary, as demonstrated by set theory, which allows an ordered set to be constructed from an unordered set. This means arbitrary relations can be assigned to a group of things with no relations.

An "unordered set", a group of things which have no order, is really an incoherent fiction, an impossible situation, because things must have position. So mathematics clearly does not give us a true representation of the reality of objects.

Such as reflecting the positive number line over the origin and reversing the sign of the reflected numbers. In other words, positive and negative numbers are opposite numbers. — Andrew M

As I explained, this is an incorrect description because zero is a part of the number line. If zero was not a part of the line, we could say there is two distinct lines, as you seem to be implying, negative and positive lines, one the reflection of the other. But that is not the case. What we have is one line, of which zero is a part. The existence of zero, as a number, means that numbers do not have an opposite number. If numbers have an opposite, what is the opposite of zero?

This user has been deleted and all their posts removed. — Deleted User

That's an excellent username. And what better way to admit that you were wrong, then to delete all your posts.

A particular apple is a collection of its parts. Is the apple not an object? What is an object then? — litewave

An object is much more than a collection of parts. Each different object has its parts ordered in a particular way. It is the order of the parts which creates the unity which you seem to want to call a collection. A collection with no parts (if this could be in some way coherent) has no order, therefore cannot be an object.

Do you have a link to a definition? — Andrew M

OED: invert: reverse the position, order or place of.

but he knows what he knows. — Real Gone Cat

Thank you.

Obviously, negatives are not treated as the direct inverse of positives, because two positives multiplied together produce a positive number, and the two negatives multiplied together also produce the same positive number.

An empty collection is a collection of no parts. A non-composite object. — litewave

Litewave, a collection is not an object. Therefore an empty collection is not a non-composite object.

They are particulars located in space and time. — litewave

They have no location, that's the issue with quantum uncertainty.

Because the recipe describes relations between particular circles, like translation, rotation, scaling. — litewave

No, the recipe for making a circle, which you produced, does not describe relations between particular circles.

Resemblance comes in various degrees and you can understand sameness as maximum or exact resemblance. So the meaning of resemblance also covers sameness. — litewave

Exact resemblance is incoherent, for the reasons you described. If there is supposed to be no difference between two things, they cannot be assumed to be two things, they must be one and the same thing.

See the mathematical definition below. — Andrew M

That's the "additive inverse". It does not mean that negative numbers are the inverse of positive numbers in a general sense, only in the operation of addition. Without that qualification it wouldn't make sense to say that a thing (zero) could be the inverse of itself, because there would be no inversion involved there.

There are also two distinct conventions for natural numbers and integers (which include negative numbers). With integers, a larger number can be subtracted from a smaller number. With natural numbers, it can't. — Andrew M

Yes, distinct conventions for numbers is a real issue, which I take as evidence against Platonism. How could a number be a single object, if there are different conventions for meaning?

With complex numbers, the negative is still the inverse of the positive. — Andrew M

But negatives are not the inverse of positives, that's the point, and it's what the fact that there is not a square root of a negative number indicates. The problem is that zero occupies a position on the number line. If it was a simple inversion, the count would go from one to negative one, as the two directions would be the inverse of each other. But there are two spaces between one and negative one. So zero occupies a place in the count, it plays a real role, and this is why the negatives are not a simple inversion of the positives, because that would rule out a position for zero. And it is also the way that we conceive of zero, as a divisor between the haves (positive) and the have nots (negatives), that makes us say that two negatives multiplied together must make a positive, but we do not say that two positives multiplied together must make a negative.

I don't think so. It remains true that negatives do not have *real* square roots, and that's the same as saying that if your domain is discourse is restricted to real numbers they have *no* square roots. The complex plane is a perfectly natural extension of the real line. — Srap Tasmaner

I don't follow this. If, within the domain of real numbers, negatives do not have square roots, then the complex plane, within which negatives do have a square root, is outside the real line, something different from it, and not an extension of it.

It would be a concrete entity without parts. — litewave

No, it would not. It would be a collection of parts without any parts. That's what your statement was, "empty collections". Your assumption that this could constitute a concrete entity is unfounded, because concrete entities as we know them actually have parts. The appeal to fundamental particles does not help you because they are obviously not known as concrete entities.

Mathematics is full of infinities and it doesn't mean that it is unsound although infinities can be pretty mind-boggling. — litewave

I've argued in numerous places on this forum that such mathematics actually is unsound. Soundness consists of truthfulness, and pure mathematics has no respect for truthfulness. So...

They are two different particulars that are the same in the way that they are red. — litewave

But "same" is the relationship which a thing has with itself. So if two distinct things are "the same" with respect to being red, then the concept of "red" cannot be a resemblance relation, which is a relationship of similarity, it would be that the two things both partake in one and the same thing, the concept "red".

When two objects are the same it means that they are also different in some way because if they were the same in every way then they would be one object and not two. — litewave

You are not respecting the law of identity. Two distinct things cannot be said to be the same, as you suppose here. If they are said to be "the same", then they are said to be one object not two. "Same" is reserved for the relationship a thing has with itself. So you are talking about being the same, in one specific way.

They are two different particulars that are the same in the way that they are red. — litewave

As I explained above, if they are the same with respect to being red, then being red means the very same thing for each of them, and this cannot be construed as a resemblance relation, which would imply that they are similar with respect to being red, not the same. If they are the same with respect to being red, then we might say that they both partake in one and the same thing, the concept red.

Ok but for example, what is the underlying thing that underlies all circles? One thing is clear: it does not look like a circle at all because if it looked like a circle it would be a particular circle and not a universal one. A particular circle is continuous in space but a universal circle would not be because it is not supposed to be located in some continuous area of space. A universal circle looks more like a recipe how to create all possible circles from an arbitrary particular circle: first define a particular circle by specifying all points on a plane that are the same particular distance from a particular point and then create additional objects by translating, rotating or scaling (enlarging/shrinking without deformation) this particular circle and you can call all those additional objects "circle" too. And they all resemble each other in the way of being a circle because any of them can be mapped onto any other via the relation of translation, rotation or scaling, and no other object can. — litewave

If it is the case, that "A universal circle looks more like a recipe how to create all possible circles", then I do not see why you want to describe this as a resemblance relation. A recipe, blueprint, or whatever you want to call it, in no way states a resemblance relation. And even if the blueprint, or production instructions end up producing similar things, this does not imply that the production instructions state a resemblance relation. The instructions make one set of statements, which if followed in action numerous times, will produce a number of similar things.

Yes, that's how I think each particular is constructed. Except that there may be empty collections (non-composite particulars) at the bottom instead of infinite regress. But even if there was infinite regress I am not sure that would be a problem, as long as the whole (infinite) structure was logically consistent. — litewave

How could there be a concrete entity which is an empty collection of parts? That makes no sense logically, an infinite number of zeros does not make one. The issue with infinite regress, is not that it is not logically consistent, because maintaining logical consistency with unsound premises is what produces infinite regress. So the appearance of infinite regress is an indication of unsound premises. This is because the result of infinite regress is that it renders the thing described by the unsound premises as unintelligible due to the infinite regress. Therefore such premises must be considered irrational because they presuppose that the thing to be understood cannot be understood because an infinite regress (therefore unintelligibility) is accepted as the truth, instead.

Yes, they have a different location and thus different relations to the rest of reality, which makes them two different particulars which however resemble in the sense that they are red. — litewave

That they are both the same, with the same name "red" is an unjustified conclusion under this description. As you yourself say the instance of colour here is a distinct particular from the instance of colour over there. So the proper logical conclusion is that it is incorrect to say that they are both the same colour, you have stipulated that they are different. The idea that there is a resemblance relation between them just comes about from your refusal to accept the true conclusion that they are completely distinct. You deny the reality of the logic, that if they are not the same, they must be different, and so you propose some sort of compromised sameness "resemblance" instead. But this principle is not supported by empirical evidence, nor logic, it's just a product of your denial, a sort of rationalizing, which is really irrational.

There is an underlying sameness but I am not sure that there would need to be a single object (universal) to "produce" the resembling particulars. — litewave

I think you need to respect the meaning of "same" as described by the law of identity. "Same" means one and the same, "a single object". "Similar" has a completely different meaning, as it implies distinct things, rather than one thing, as "same" does. So if there is an "underlying sameness", this means that there is one and the same thing which underlies the two distinct instances, such that they are multiple occurrences of the same thing, just like two distinct occurrences of "now" could be said to have the same underlying thing, time. But two similar things do not require any underlying sameness, just a judgement of "similar", which could be based in any sort of assumption. If we say that two instances of "now" are similar, rather than having an underlying "time" which makes them two instances of the same thing, then we might employ any arbitrary principle whereby we would say that they are "similar". But this judgement of "similar" is completely arbitrary.

The math was entirely adequate but there was no natural picture, hence a lack of understanding. However, if negative numbers are thought of as the inverse of positive numbers, then they can be visualized. For example, credits and debits in banking. Or walking forwards and backwards. — Andrew M

The issue of imaginary numbers is different though. It is an issue of there being two distinct conventions, yet each convention is correct in its own field of application. In the one case there is no square root of a negative number, in the other case there is. This means that there is two completely distinct ways of conceiving negative numbers, and not a simple matter of negative being the inverse of positive. It is how the negative are conceived to relate to the positive, that creates the problem, i.e. it is not a straight forward inversion due to the role that zero plays.

Odd that it needed saying, but well said. — Banno

The speaker might know that the book is in the car but still be literally honest and correct, in saying "The book might be in the car". — TonesInDeepFreeze

If that's what you call "honest" communication, and "well said", then it's no wonder that I have an aversion toward communion with you two. Your principles for sharing with me are not up to the level of my principles for sharing with you. Sad but true. What a shame.

Ok, but I am saying that these "universal principles" are just resemblance relations between particulars rather than additional entities (universals) that instantiate in the particulars. — litewave

But in saying that, you are already assuming the existence of particulars. And by taking the existence of particulars for granted, you do not consider the process whereby we individuate particulars from the universe, in your representation, and this skews your perspective.

If you step back and take a better look, you'll see that the real problem is the question of individuation, by what principle do we say that this is a separate entity from that, as a particular, or individual. If you do this, then you'll see what was evident to the ancients whom I mentioned, that the universal is necessarily prior to the particular. Therefore your whole question, or starting point, as the issue of how we produce universals from particulars is based in a complete misunderstanding of reality.

How? — litewave

You don't see how this produces an infinite regress? If a concrete particular is a collection of concrete particulars, then each concrete particular in that collection is itself a collection of concrete particulars, and each concrete particular in that collection is itself a collection of concrete particulars, ad infinitum.

t seems that I could in principle define a part of the ball that constitutes the ball's particular red color. — litewave

See, look what you are doing here. You are individuating, separating out "a part of the ball", and passing judgement, to make it into a particular thing which you can refer to. But at the same time, you want to take it for granted that particulars have already been individuated, and we proceed from those particulars to produce universals.

Once you accept that a particular is produced from this sort of individuation, then you must see that the way that a human being produces universals is completely dependent on the way that one produces individuals. So we cannot proceed toward understanding how one produces universals, unless we first produce an understanding of how one produces individuals. If we take the existence of individuals for granted we cannot get anywhere.

So, for example there is a resemblance relation between two red particulars in the sense that they are both red. — litewave

So, let me explain, using this example. If the two supposed "particulars" are both the same in the sense of red, then why would you say that they are two, and not one instance of "red". If they are different shades of red, or something like that, then there is nothing to support saying that they are both the same colour, red. If they appear to be the exact same colour, then whatever it is which separates them as two distinct particulars, must be something other than colour. But how could it be that two exactly the same instances of colour could come to exist under completely distinct circumstance? Wouldn't they have been originally one thing which got divided? Any way you look at it, we would have to conclude that there is something "the same" about the circumstances, something underlying, which is truly the same, which could produce the exact same colour in two completely different situations. And if we say that the colour is not really exactly the same, it is only similar, then we have no reason to say that they are both the same colour, "red". Therefore we must conclude an underlying sameness as the reason why they are both red, or else saying that they are both the same colour, "red", is completely unjustified.

(Thanks for the notes on the ancients, btw.) — Srap Tasmaner

I appreciate someone who is receptive to different perspectives. I think that's what philosophy is made of.

Now we might think — identity of indiscernibles to the rescue! And now that we come to it, how did we imagine the sort of partial particular I described being a numerically distinct entity? It's not, after all; it's only an aspect of a 'genuine' concrete entity. Not even a part of it, but something that, obviously it seems, cannot exist on its own, but only as an aspect of something concrete.

No problem; we knew that as soon as we said we were creating an abstract object (the red of this ball) from a concrete object (this ball). But if it's no real objection that these things can't exist on their own, then we can't rely on their individual existence to underwrite their being numerically distinct. Maybe abstract objects can be numerically distinct, but if they can it's not the way regular concrete objects are. — Srap Tasmaner

Actually, I would say that the partial particular, for example the particular redness of this ball, is a concrete part of the concrete whole (this ball). A concrete object is structurally a collection of other concrete objects and there are various overlapping collections inside this collection. In the case of this ball, one of those overlapping collections is a particular red color because the structure of that collection is such that it reflects certain wavelengths of incoming light. — litewave

The problem with litewave's representation here is that the existence of the particular "concrete whole" is taken for granted. Srap demonstrates how this is not an acceptable starting place. The idea that we build universals through observation and abstraction from particulars, is just not consistent with what we really do. Abstract, "pure mathematics" shows that we dream up universal principles (axioms) first, from the imagination, or they come to us intuitively, then we try to force the particulars of specific circumstances to be consistent with the universals. If we cannot produce such consistency, the universals get rejected and replaced. What is neglected, or left out from litewave's representation, is that whenever we proceed toward comparing particulars, we do so with a preconceived standard, or rule, for comparison. A comparison without such a standard is impossible, therefore the standard must be prior to the comparison and cannot be properly represented as being produced from it.

This is the problem which Plato faced with Pythagorean Idealism, the question of how the reality of the particular individual, the "concrete object" is supported, justified, or substantiated. Litewave's suggestion, that a concrete particular is a collection of concrete particulars had already been demonstrated to be faulty because it was known to produce an infinite regress. The Idealists proposed that the existence of the particular is supported by the universal, the Idea, and this was seen to be necessary from the reality of the concept of "generation". When a particular being comes into existence, it is necessarily the type of being which it is, therefore the universal, or Idea, must precede in time, as a cause of existence of the particular. The universal must precede in time, the particular, in order for the particular to be caused to be the type of thing which it is.

The problem which Plato exposed is that the Pythagoreans supported their Idealism with the theory of participation, and this could not account for a causal relationship between the universal and the particular. A particular concrete entity is supposed to be the type of thing which it is, through the means of participating in the Idea. So Plato showed how, in this representation, the Idea is passive, while the particular thing is active, by actually participating in the Idea, and this cannot account for causation. Then, in "The Timaeus" he proposed an alternative whereby the Idea is actual, and acts to cause the reality of a particular concrete entity being the type of thing which it is, therefore the Idea acts to cause the existence of the particular thing.

I think that a general property without particular instances is an oxymoron because it is inherent in the meaning of "general" property that it is instantiated in "particular" instances. — litewave

Again, this is an example of your misrepresentation. We can and do imagine many general properties without any particular instances. That's obvious in mathematics.

The beauty, if I could call it that, is this: if the potential energy of a rock is 6 Joules, what it does/can do is fully accounted for by these 6 Joules it reportedly possesses. — Agent Smith

As I said, there is no equivalence, due to entropy, which is the supposed "energy" which is unavailable, neither potential nor kinetic.

Potential energy is simply stored energy we can tap. The word "potential" isn't to be understood philosophically, as antipodal to actual (vide Aristotle). What sayest thou? Just a poor choice of words, a misnomer, or a clue that something's not quite right? — Agent Smith

Potential energy is energy relative to a thing's position. Kinetic energy is energy relative to a thing's activity. Clearly there is a very substantial difference between kinetic energy and potential energy.

Usually the concept of work relates to a change of energy, kinetic or potential. When an object follows a path through a force field, if that field is conservative, the path the object takes from point A to point B is immaterial regarding work; all such paths produce the same work. This idea aligns with Cauchy's Theorem in complex analysis. — jgill

This may be true of potential energy, but since kinetic energy relates to the specific activity itself, we cannot say that the path the object takes is immaterial. And this is why potential energy and kinetic energy are essentially non-convertible, due to unaccountable losses like friction, etc., what is called entropy, so we have no perpetual motion. But the law of conservation might pretend that they are convertible.

I don't know if I agree with this. If I had a mechanical clock with a spring windup mechanism and it was fully wound, I would say the potential energy was within the clock, and in particular, within the wound spring. I wouldn't suggest it was floating within the clock or that it was somehow extractable from the spring so that it could exist separate and apart from the spring. — Hanover

If you followed what Frank said, you might start to see why I said "potential energy" really doesn't make any sense logically.

Consider, the definition of energy stated by , "the capacity to do work". As a "capacity", this means energy is fundamentally a type of potential. Now we qualify that with "potential", and we have the potential for a potential. Of course two potentials don't make something actual, but what does it make? What sense can you make of 'the potential for a capacity (potential)'? Is it the possibility of a possibility? But let's start simple, what type of existence can a potential be said to have, in the first place?

It's gravity, but gravity is understood as the property of another object. If you remove the other object which the gravity is the property of, then you have energy as an independent entity, existing as a property of space.

As per my high school physics sutra, energy is the capacity to do work. — Agent Smith

Now consider the difference between kinetic energy and potential energy. The former would be actually having the capacity to do work, and the latter would be having the potential to have the capacity to do work. The concept of "potential enrgy" really doesn't make any sense logically, but the use of it is what gives rise to the issue points us toward, where energy is seen as an entity in itself, rather than the property of an active object. When a thing has potential energy, that energy can only be understood as the property of something else. But it's easier just to ignore the requirement of something else, allowing the energy to exist as an abstract entity.

Anyway, I thought your argument was about accidental properties, not essential properties; that we might be wrong about the ball being red, not that we might be wrong about the ball being a ball. I don't see how the community could be wrong about what we call a "ball". Then again, I don't see how the community could be wrong about what we call "red". — Luke

I was talking about the reasons why we say that the ball is a "red ball", and we assume that it cannot not be red. because it is a red ball I think this is analogous to the reasons why we used to say that Pluto is a planet, and we would have been inclined, at that time, to say that it cannot not be a planet.

What object are you talking about here? — Luke

I don't see the relevance.

Yes, there is reasoning involved in teaching language. My point was that in teaching the meaning of the word "red" to someone, the teacher doesn't arrive at the meaning through "reasoning". The teacher knows how to use the word; they must, otherwise they couldn't teach it to someone. Recall that this was in response to your statement: — Luke

I don't see the relevance.

The teacher does not "impose on the ball that it cannot not be red, just because our reasoning says so." The teacher and other English speakers call it "red" because that's what we call it. — Luke

That's not what the issue was though. The issue was whether this particular ball which we classified as "red ball", because we thought it was that type of ball which could not be other than red, would still be the same object, this ball, if somehow it became apprehended as not red. We called it "red ball" because we thought it is necessarily red. But if it is demonstrated not to be a red ball, like Pluto was demonstrated not to be a planet, then we ought to accept that the reasoning by which we identified it that way was wrong, and not try to impose on the ball that it must be a red ball.

Teachers, and other English speakers called Pluto a planet, because that's what we called it. When the reasoning was demonstrated as faulty, these English speakers had to adjust. They did not insist that Pluto must be a planet because that's what we call it.

I just don't think that makes our purpose constitutive of the objects we interact with. I think they have to be there, as they are, for us to have the options we do, among which we select the one that aligns with our purpose. If you can sometimes sort papers by author and sometimes by keyword, depending on your purpose at the moment, it's because they have authors and keywords. If they didn't, these wouldn't be options for you. — Srap Tasmaner

It might be useful for you to reconsider this to some degree. We, as living human beings, have sensory systems which have evolved from nothing. That means that over the millions or billions of years of evolution which have produced our sensory systems, the sensory systems have been shaped and formed by what has proven to be useful. So the way that we perceive things, as objects, is a product of that usefulness. The important thing to note, is that unlike your example, alternative options for how we perceive things, are not there for us.

We have been forced into this mode of perceiving because it served some evolutionary purpose, and now it is a fixed part of our being, which we cannot opt out of. And as you mentioned in the other post, we just keep getting pesky scientific refutations. Science has provided us the means to get beyond the limitations of our sensory equipment. Plato's principal message was that the senses deceive us in our quest for truth, follow the intellect not the senses. And I suppose, through the presupposition of free will, we've managed to develop the intellect as an alternative option, under the notion that it can operate independently from the senses. In Aristotle's ethics, contemplation is the highest virtue.