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

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

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

An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality. — aletheist

You don't see this as a problem? Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. What could this possibly mean, other than something contradictory? It doesn't have any measurable colour but it has real colour.

Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit. — aletheist

But a line is specifically one dimensional, and an infinitesimal is not. So if you zoomed in on an infinitesimal why would you see it as a one dimensional line rather than as three dimensional, four dimensional, or even an infinity of dimensions for that matter? If it might be an infinity of dimensions, then the purpose of the infinitesimal is self-defeating.

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

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

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

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

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

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

You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? What nonsense is that? Mathematical objects exist as specific definite things. That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. That's why the attempt by speculative logicians and mathematicians to bring "infinite" into the realm of mathematical objects is doomed to failure as inherently contradictory.

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

Each of those shades of colour is measurable though. You have defined the infinitesimal as having an immeasurable dimensionality, yet still having a dimensionality. Since dimensionality constitutes being measurable, this is like saying that infinitesimals have something measurable (dimensionality), which cannot be measured. That's blatant contradiction.

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

The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal".

You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? — Metaphysician Undercover

An infinitesimal is not a number.

That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. — Metaphysician Undercover

What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else.

Each of those shades of colour is measurable though. — Metaphysician Undercover

False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum.

Since dimensionality constitutes being measurable ... — Metaphysician Undercover

It straightforwardly begs the question to define dimensionality as "being measurable," when what is at issue is the logical (not actual) possibility of dimensionality that is not measurable. Measurement entails discreteness, but we are talking about true continuity.

The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal". — Metaphysician Undercover

By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero.

What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else. — aletheist

How is it that mathematics is necessary for building things, yet numbers do not interact with anything? That's the nonsensical claim, that numbers do not interact with anything. I suppose engineering could be done without numbers? And if numbers are necessary, how so if they don't interact with anything? Obviously numbers interact with things or else they could not be necessary for building things.

False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum. — aletheist

Of course they're not measurable shades if they're not actual shades, only potential shades. They are simply imaginary, so of course they cannot be measured. Is this how you conceive of the continuum as well? Is it simply imaginary as your example seems to indicate? I think it's purely imaginary, don't you?

By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero. — aletheist

This is what is nonsense. Your one-dimensional infinitesimal is just a short line. You arbitrarily claim that its length is less than any assignable value, but there is no such limit to our capacity to assign a length value because numbers are infinite. So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit.

How is it that mathematics is necessary for building things, yet numbers do not interact with anything? — Metaphysician Undercover

Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.

I suppose engineering could be done without numbers? — Metaphysician Undercover

I suppose it depends on how you define "engineering." At this stage of my own career as a structural engineer, I spend most of my time making high-level decisions that involve the exercise of practical judgment obtained through experience, rather than crunching numbers.

Obviously numbers interact with things ... — Metaphysician Undercover

Really? Where can I find a number so that I may interact with it?

Of course they're not measurable shades if they're not actual shades, only potential shades. — Metaphysician Undercover

I have consistently characterized a continuum and an infinitesimal as real--that which is as it is, regardless of what any individual mind or finite group of minds thinks about it--but not actual.

So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit. — Metaphysician Undercover

This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.

people will readily admit that the real between 0 and 1 are infinite despite — MindForged

Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED.

Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity).

When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua.

If its not a quantity, which it is not by definition, we should not assign it to physical quantities like, time, size, mass etc...

Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary. — aletheist

Actually you seem to have misunderstood what I meant. I didn't mean mathematics is necessary for building all things, but for some things. So my argument remains the same. Of these things which mathematics is necessary to build, the mathematics must somehow interact with things in order that these things get built.

Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary. — aletheist

I don't see how my computer could have been built without mathematics. Regardless, let's just say that mathematics is useful for building things, as you say. How could mathematics be useful in building things unless it somehow interacted with things? You might say that the human being is a medium between the thing built and the mathematics, but the human being is also a thing, and the mathematics must interact with that thing in order for it to build the things which the mathematics is useful for. So the mathematics still interacts with things, even though the human being, as a thing is a medium between the mathematics and the thing built..

This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real. — aletheist

I never made any such claim so instead of addressing my concerns you are just changing the subject.

... the mathematics must somehow interact with things in order that these things get built. — Metaphysician Undercover

Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists.

But the only sense in which "an infinity" is bounded is by the terms of its definition. All infinites which we speak of are bounded by the context in which the word is used. If someone mentions an infinity of a particular item, then the infinity is bounded, defined as consisting of only this item. Likewise if we are talking about an infinity of real numbers between 0 and 1, the infinity is bounded, limited by those terms. However, we are not discussing particular infinities here, which may be understood as particular (though imaginary) objects, we are discussing the concept of "infinite". — Metaphysician Undercover

This is mistaken. My point is simple. Infinity is often intuitively understood as unbounded quantitatively. In other words, given any arbitrary number N there is some number N+1 that can be accessed from N. No set end point, basically. However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. So clearly the colloquial understanding of this infinite is not consistent.

This is false. Anytime "infinite" is used to refer to something boundless, or endless, it refers to something made up by the mind, something imaginary or conceptual. We do not ever observe with our senses anything which is boundless or endless, because the capacities of our senses are limited and could not observe such a thing. Since the capacities of our senses are finite we know that anything which is said to be infinite is a creation of our minds, it is conceptual, ideal. — Metaphysician Undercover

I didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? All that tells us is that those observations are finite and so the next ones made will likely be finite. It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible. We have the math to make perfectly logical sense of this and all current physics assumes this is the case (even if you wanted to suggest loop quantum gravity, I could suggest the equally speculative string theory where space can be infinitely divided again).

Spacetime is conceptual. This is the problem I had with your last post, you reified "space", making it into some sort of an object to justify your position. In reality, "space" is purely conceptual. We do not sense space at all, anywhere, it is a constructed concept which helps us to understand the world we live in. Furthermore, "infinitely divisible" is an imaginary activity, purely conceptual. We never observe anything being infinitely divided, we simply assume, in our minds, that something has the potential to be thus divided. — Metaphysician Undercover

I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object.

I never defined "potentiality" as ineffable. It may appear to you that potentiality is contradictory ifyou do not understand the concept, but Aristotle was very specific and explicit in his description of what the term refers to, — Metaphysician Undercover

Whatever Aristotle may have said, refer to what you said before:

The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to.

You said it's impossible for humans to understand and yet clearly you think you can explain what about it makes it impossible for humans to understand. So unless you can explain something about things you can't possibly understand it sounds like you're contradicting yourself.

The logicians at the time decided that the best way to proceed was to change the premises, the defining terms of "infinite". What I am arguing is that misunderstanding is not due to faulty premises, but to faulty logical process. Zeno's paradoxes deceive the logician through means such as ambiguity or equivocation, by failing to properly differentiate between whether the aspects of reality referred to by the words, have actual, or potential existence. That's what Aristotle argued. So the logician gets confused by a conflation of actual problems and potential problems, which require different types of logic to resolve, and are resolved in different ways. Instead of disentangling the potential from the actual, the logicians took the easy route, which was to redefine the premises. All this does is to bury the problem deeper in a mass of confusion. — Metaphysician Undercover

You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have.

You haven't addressed the issue here. You only support these claims with a reified "space", assuming that space is a physical object to be studied, and not a conceptual object. — Metaphysician Undercover

I never said space was physical. An object, sure. It has properties after all and we have studied these properties. I'll come to this in a moment.

What's this then? — Metaphysician Undercover

, you are treating "space" as if it is something described by geometry. In reality, since we can use various different geometries to describe the various types of objects we sense, there is no such thing as "space". We might be able produce a concept of "space" from this geometry, and another concept of "space" from this other geometry, but it really makes no sense to talk about "how space is", or "if space is curved...", because there is no such thing as "space", not even as a concept. — Metaphysician Undercover

I don't really see how you are saying anything here because you're moving between unrelated points. When I referred to "how space is" I was talking about the actual structure of spacetime, not the more vague, general concept of space. Some abstract geometric spaces are curved and some are not. Whether or not the structure of the actual space of the universe falls into one or the other is a physics question, and current physicsand observational evidence says space is curved. If you cannot even accept this there's no point in this.

This is why your geometrical examples are irrelevant, and way off the mark. You are talking about geometry as if it is created to describe some sort of "space" — Metaphysician Undercover

Er, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you).

However, this is totally uncalled for. We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects — Metaphysician Undercover

I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me?

Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED. — Devans99

You do not understand the concept of cardinality, do you? The size of the interval is the size of the continuum, aleph-1.

Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity). — Devans99

Why do you keep asserting this as fact? That is not how "infinity" is defined in mathematics because it's too hazy and informal a definition. An infinite set is, e.g. the transfinite Cardinals, a set whose members can be put into a one-on-one correspondence with a proper subset of themselves. There is absolutely no mention of being "greater than any assignable quantity, you're just wrong. Find one mathematics textbook that formally defines and describes infinity that way. Go on, I'm sure you can do it... (Obviously I'm being sarcastic here).

When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua. — Devans99

That's not an argument, that's a statement that you cannot justify. As it happens, it's perfectly understandable why the CARDINALITY doesn't change. The set does change if you add a new unique element, but the size cannot change by mere finite additions because we can still world new numbers to put into a function with the new element that was added.

I was just helping a friend out with a website. The RGB color values could be changed by the file in question as needed, but there seemed to be an oddity. Because color values are cappped at 255 per channel, adding any number to that channel resulted in no change to the value of that channel. I suppose 255 must not be a quantity then since adding to it did not cause the value to change. Saturation mathematics must be incoherent despite it's use in computer science then!

It's a little funny that mathematics by unjustified dogmatic assertion went out of vogue for everyone besides the ultrafinitists.

Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists. — aletheist

You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true.

However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. — MindForged

Actually, that is what is mistaken. In the case of the reals between 0 and 1, it is quite obvious that 1, like 0, is a defining point, and therefore a beginning point rather than an end point. What is claimed is that there is an endless number of points between 0 and 1. It is impossible that there could be a direction of procedure, because if we started at zero, and tried to progress toward 1 as an ending point, it is impossible to name the first real number after zero. Any named number after 0 would have more numbers between it and 0, and we'd have to turn around and go back, heading away from 1. Therefore, in this instance it is false to represent 1 as an "end point". There are two defined start points, 0 and 1, with an infinity of points between them. No end point.

didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? — MindForged

I guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite.

It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. — MindForged

I know, I believe I already described to you how definitions are arbitrary. But those definitions which demonstrate a true correspondence are considered to be true definitions. So if we observe that the clear sky is always a similar colour, and we name this colour as "blue", so that everyone calls this colour blue, then we can define "blue" as "the colour of the sky". There is true correspondence because that is how people use the word "blue". But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence.

Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible — MindForged

Why is this "worse"? Time and space are purely conceptual, just like numbers. Numbers are conceptual and infinite. If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? A conclusion reflects its premises. The premise is that numbers are infinite. If time and space are concepts created from numbers they will reflect this infinity. Unless we allow that time and space are concepts created by something other than mathematics, they will necessarily be infinite. If time and space are other than mathematical, what would be the basis of these concepts, observation? Observations are necessarily finite. Therefore we have an incompatibility between the concepts of space and time which are consistent with mathematics, and the concepts of space and time which are consistent with observations. This has manifested as Zeno's paradoxes.

I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object. — MindForged

The problem is that things unobserved do not enter into conceptions produced from observations So, even if there is a real thing out there, like space or time, which is truly infinite, the limitations of our senses deny us the capacity to observe the infinity of this thing. That's the classical, or colloquial understanding of "infinite", that it's impossible for the human being to observe. Let's assume that space and time are infinite, as the mathematical conceptions tell us, but our observations are incapable of corroborating this due to the limitations of our senses. Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? Do we face the fact that our observations are limited, and therefore fail us in this realm, and maintain a pure infinite in our concepts of space and time, or do we denigrate the pure infinite concept, and produce a new concept of "infinite" which is more consistent with our faulty observations? The latter is what the logicians have done, and what you seem to insist was the right thing.

You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. — MindForged

What you're not respecting, is that for Aristotle ideas are part of reality. He was a student of Plato and was well trained in an ontology that holds ideas as real. To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. Infinity, as well as mathematical ideas are very real for both Plato and Aristotle, so placing "infinity" into the category of ideal is not partitioning it away from reality.

After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have. — MindForged

You seem to be missing the point. I agree with what you have described here, a possibility is defined by actuality, what actually is. This is a specific possibility, it is only correctly "a possibility" if the actuality permits, otherwise it's impossible. Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible?

I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me? — MindForged

Yes, I see why my view of geometry makes no sense to you. You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that?

You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true. — Metaphysician Undercover

As usual, this reflects conflation of the real with the actual.

Numbers are conceptual and infinite. — Metaphysician Undercover

If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?

Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? — Metaphysician Undercover

Recognize that continuous motion through space-time is a more fundamental reality than discrete positions in space and/or discrete instants in time. We arbitrarily impose the latter for the sake of measurement and calculation.

To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. — Metaphysician Undercover

Where on earth have I ever suggested that ideas are not real? Perhaps this reflects yet another conflation, this time between two definitions of "idea"--the content of an actual thought vs. anything whose mode of being is its mere possibility of representation. The latter is real even if it never actually gets represented, which means ...

I agree with what you have described here, a possibility is defined by actuality, what actually is. — Metaphysician Undercover

... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible.

If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right? — aletheist

"Infinite" is a description of the numbers, as such it is qualitative, not quantitative. Mathematics cannot deal with the concept of "infinite" which is a description of mathematical objects made from outside the principles of mathematics, because it is not a mathematical principle. That's the problem here. The point being that whatever category you put the mathematical objects into, the descriptive term "infinite" is of another category, as the difference between the territory and the map, one being the object, the other a description of the object. The problem occurs when we attempt to make "infinite" a mathematical object.

Where on earth have I ever suggested that ideas are not real? — aletheist

You insistently claim that numbers have no actual existence. The #1 definition of "real" in my OED is actually existing. So I concluded that you do not believe numbers to be real. You insist that numbers cannot interact with things in the world. Now you claim that ideas are real. I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. What sense is there in this, to allow for something real, which cannot interact with anything else in the world? So I haven't any idea what you might mean by "real" now because you seem to be claiming that there are real things which cannot in any way interact with the world we sense.

... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible. — aletheist

I don't see how "possibility" is at all relevant in your reality. It cannot interact with the actual world, as a distinct mode of being, so how could it be relevant? And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed.

I am afraid that I cannot make heads or tails of your first paragraph because of the confusion it exhibits regarding the meaning of terms--infinite, qualitative, quantitative, mathematics/mathematical, category.

You insistently claim that numbers have no actual existence. — Metaphysician Undercover

Yes, but I claim just as insistently that numbers are nevertheless real, because ...

I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. — Metaphysician Undercover

I have stated this explicitly and repeatedly--I deny that reality and actuality/existence are synonymous. Reality consists of that which is as it is regardless of what any individual mind or finite group of minds thinks about it. Actuality/existence is that which reacts with other like things in the environment. Reality includes some possibilities and some conditional necessities that may or may not ever be actualized.

And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed. — Metaphysician Undercover

There is no contradiction. Something is logically possible if it is merely capable of representation; something is actually possible only if it is also capable of actualization.

There are two defined start points, 0 and 1, with an infinity of points between them. No end point. — Metaphysician Undercover

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. The only way your argument could work would be by the hilarious assumption that the quantity of real numbers between any arbitrary interval were finite. Defining the start and end of something does not mean that end is not an end point. For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over.

guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite. — Metaphysician Undercover

And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning.

But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence. — Metaphysician Undercover

The problem is that is not the exclusive colloquial definition of infinite for reasons I've already mentioned.

If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? — Metaphysician Undercover

What you seem to be missing is that the math is used to model the world and so far no model of finite space or time has any particular empirical or theoretical backing. You'd have to pin your hopes on something like Loop Quantum Gravity, but that's highly speculative and has about as going for it currently as String Theory does (not to say it won't change), whereas time and space are still modelled as continuums in both quantum mechanics and relativity. If the current models are accepted, it's just a performative contradiction to give them credence but to arbitrarily say some of their fundamental assumptions are to be presumptively excluded from reality.

Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility — Metaphysician Undercover

This is just false. Mathematics already resolved Zeno's paradoxes so clearly adopting the mathematical models of spacetime does not create paradoxes given we know how to resolve the apparent issues Zeno saw with having them be infinitely divisible. Zeno made fundamentally mistaken assumptions about the consequences of trying to cross an infinite series, they turned out to be negated.

What you're not respecting, is that for Aristotle ideas are part of reality. — Metaphysician Undercover

Word game. By "reality" I meant being actual. You've already said you don't think this is possiblefor infinity, I was showing you how you were holding a hypocritical standard that only applies when other people use definitions you don't like, but you're perfectly find doing it yourself even if it's not the colloquial, "true" definition.

Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible? — Metaphysician Undercover

Contingency and possibility are not the same thing. Necessary truths are also possible truths (because possibility just means truth in at least one possible world). Contingency means some modal proposition is true in some worlds and false in others. That aside, I don't see the relevancy in your questions about modality. In nearly all cases, potential is just a synonym for the term "possible". E.g. Every English speaker can readily understand "I have the potential to be a doctor", but sentences like "I have the potential to be a doctor but I cannot be a doctor" have to be disambiguated since it switches between two different types of modality (logical possibility and physical possibility), otherwise it's a flat contradiction. You can consistently say "I have the potential to be a doctor but in actuality I functionally cannot".

If you say X is a potential infinity but it cannot be actualized you are either contradicting yourself or you're switching between 2 different modalities. If it's the former, well that's not workable on pain of absurdity. If X cannot be actualized it's not a potential anything, the label doesn't fit. If it's the latter, then you're playing a shell game. You have to argue that Infinity isn't metaphysically possible, it's not contradictory so it's not inherently off the table for a consistency issue.

You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that? — Metaphysician Undercover

Dude, just prior to the part you quoted I said that the canonical application of geometry was for measurement:

, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you). — MindForged

My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing.

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. — MindForged

Right, you think that there could be an "end point" without an order. You really like to argue by way of contradiction, don't you?

For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over. — MindForged

Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion.

And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning. — MindForged

Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question.

So, what kind of infinite thing (infinity) do you think you could observe?

Word game. By "reality" I meant being actual. You've already said you don't think this is possible, — MindForged

That's what I meant, "actual". If you saw my discussion with aletheist, you'd see that. For Aristotle, ideas, concepts, have actual existence, actualized by the human mind. This is the argument he uses against Platonic idealism. Ideas cannot be eternal, because only actual things can be eternal, and ideas are only given actual existence by the human mind, so they have a beginning and are therefore not eternal.

My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing. — MindForged

More of the same, nonsense. The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter.

Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion. — Metaphysician Undercover

I didn't assert a contradiction. Your claim was that you have to be able to count the series in order to declare an end point, which is false. A race can be run backwards, it can be run from the middle out to either end, etc. The order is irrelevant. A race doesn't end at a random point, the end is defined when the beginning is defined.

Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question. — Metaphysician Undercover

Disingenuous. The point is you cannot define it as necessarily impossible and then claim to have proven it to be the case. Induction only yields probable conclusions, you claimed the conclusion that infinity was impossible to actualiz was necessarily false, and you brought up inductive generalizations to prove that. You made a non sequitur, induction cannot give you necessary conclusions.

So, what kind of infinite thing (infinity) do you think you could observe? — Metaphysician Undercover

Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible.

The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter. — Metaphysician Undercover

You say things like I can't just quote what was said before. The question was whether or not geometry was about measuring things. I said that was the canonical use of the discipline, but that's not what the discipline itself was about. That you're trying to claim the division being pointed out avoiding the issue is ridiculous. I brought it up because you said this:

We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects — Metaphysician Undercover

But that's absurd. We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. The common assumption was that reality could not be any other way than as a model of a Euclidean Space until Non-Euckidean geometry came along and Relativity gave credence to thinking our actual space was best modelled as a pseudo Riemannian space. Funnily, Euclid made as a base assumption in his geometry that space was an infinite plane but I'm sure you'll object to that without question begging and ignoring that Euclid's assumption contradicts your claim that geometrical axioms are about measurement. It's about studying abstract math structures of a certain kind.

Anyway, theory and application aren't the same and the idea that geometry is fundamentally about measurement is wrong. If you think otherwise, show where measurement appears in the formalism of common geometries.

Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible. — MindForged

Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? Our best mathematical models of buildings for structural analysis consist of finite elements, but no one would seriously claim that we observe and experience real buildings as collections of finite elements.

In any case I would suggest that space-time is an example of observable continuity, rather than observable infinity; and since the concept of infinite divisibility has proven problematic in past discussions, I would suggest infinite magnification as an alternative. No matter how much you were to "zoom in" on space-time, you would always "see" a four-dimensional continuum, all the way down to the infinitesimal level; never a discrete point at a discrete instant.

We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged

Exactly, and the same is true of mathematics in general. We generate formal hypotheses and work out their necessary consequences, only some of which turn out to be useful for measuring or otherwise analyzing actual phenomena. That is precisely why we make a distinction between "pure" and "applied" mathematics.

Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? — aletheist

I should clear this up. I posit that those models, well evidenced as they are, are the best explanation of why we make the kinds of observations we make (e.g. not reaching any sort of discrete unit of space no matter the magnification). And since both QM and Relativity have some type of continuity to spacetime, we ought to accept this until such time as we have reason not to. I don't mean I completely see the infinite totality if a thing, but that whatever credence we give to observations in establishingnor refuting the actuality of infinities, our current observations of space and time don't seem to contradict this possibility at all.

That is fair. Can you elaborate on how QM supports the continuity of space-time? What is your interpretation of the Planck length and Planck time?

To be up front, my undergrad requirements for physics didn't really get into QM, so my "knowledge" of QM is a hackjob accrued from friends, colleagues and the Google box linking papers.

All that said, I'll give it a go. From the name, many people think QM must say space and time are quantized (discrete/, but many values in QM are continuous (position for instance), and space and time are among those values. In and of itself that doesn't mean too much, since you probably could modify the theory to use discrete values for these instead (although in practice there's no benefit to doing so).

If there's a minimum distance you start messing up a lot of other things in physics, especially as it relates to Relativity currently, such as people in different reference frames measuring different Planck length due to relativistic effects. S you'd probably have to drop Lorentz Invariance, but somewhat recent experimental observations (2011) haven't borne out high enough violations of it that would be expected if spacetime were discrete at some scale:

https://www.nature.com/articles/nature08574

Planck length and time are just measurement limitations at best. It's plausible that Planck lengths are the smallest measurable lengths, but there's no current reason to thinks it's a fundamental chunk of reality. Like these are somewhat taken arbitrarily. Inagine I was measuring things based on, I don't know, the radius of the Earth. That wouldn't mean everything else should be measured as if they were a multiple or divisor of the planet's radius. Natural units are all well and good. Some make a lot of signifance out of the Planck length but all we can really say that isn't speculation is that it sets a limit on the non-negligible effects of Quantum gravity. Our models of quantum gravity would only work down to that scale, so we'd need something more to model anything that exists at a smaller scale.

Thanks--all of that is consistent with my understanding, as well. :up:

I didn't assert a contradiction. — MindForged

This is what you said:

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. — MindForged

You are saying that there can be an end point without an order of procedure. That's contradictory, "end" implies order, by definition.. Your dismissal of my argument as "nonsense" relies on the truth of this contradiction. Since it is impossible that a contradiction is true, you need to go back and address my argument properly.

We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged

Let me see if I can understand what you're saying. You're saying that we do not produce axioms in geometry for the purpose of measuring things, we do it for some other purpose, maybe just for fun, or some arbitrary, random purpose. Then, voila, it just so happens by some random chance, that the principles of geometry prove to be very useful for measuring objects. Come on, get real.

If you think otherwise, show where measurement appears in the formalism of common geometries. — MindForged

Are you serious? Measurement is everywhere in the formalism of geometry. 360 degrees in a circle is a measurement. Pythagorean theorem is a principle of measurement. I can't understand how you can appear to be so intelligent MindForged, but then fill your posts with such silly and even ridiculous statements.

It doesn't make sense to carry on this discussion because you just defend your position with contradictions and statements of random nonsense. Then you pretend that what I am saying doesn't make any sense in relation to your statements of contradiction and random nonsense.



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, I happen to know @Metaphysician Undercover (and @aletheist) from the old now-defunct philosophyforums.com.
Metaphysician Undercover tend to wander off in some direction of own makings, yet imposing own ideas on other things. :)
As far as I can tell, @Devans99 just doesn't have much familiarity with the mathematics.

As far as I can tell, Devans99 just doesn't have much familiarity with the mathematics. — jorndoe

You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time.

None of you will address this point directly.

, it's been addressed more than once by others (including here).

You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time. — Devans99

Cool. It's a a good thing "A number larger than any number" is not the definition of infinity in mathematics. The closest correct description of infinity that resembles what you're saying would be to say every infinite number is larger than any finite number. There's no logical error that results from infinity.

this reflects conflation of the real with the actual — aletheist

...and the difference between "real" and "actual" is...? :chin: