Why is that necessarily a problem? An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality.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
An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality. — aletheist
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
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?Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. — 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?But a line is specifically one dimensional, and an infinitesimal is not. — 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? — aletheist
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
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
An infinitesimal is not a number.You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? — 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.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
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.Each of those shades of colour is measurable though. — 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.Since dimensionality constitutes being measurable ... — 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.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
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
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
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
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.How is it that mathematics is necessary for building things, yet numbers do not interact with anything? — 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.I suppose engineering could be done without numbers? — Metaphysician Undercover
Really? Where can I find a number so that I may interact with it?Obviously numbers interact with things ... — 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.Of course they're not measurable shades if they're not actual shades, only potential shades. — Metaphysician Undercover
This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.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
people will readily admit that the real between 0 and 1 are infinite despite — MindForged
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
Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary. — aletheist
This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real. — aletheist
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.... the mathematics must somehow interact with things in order that these things get built. — Metaphysician Undercover
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 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
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 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
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.
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 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
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
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
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
Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED. — Devans99
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
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
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
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
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
It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. — MindForged
Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible — MindForged
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
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
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
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
As usual, this reflects conflation of the real with the actual.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
If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?Numbers are conceptual and infinite. — 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.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
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 ...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
... 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.I agree with what you have described here, a possibility is defined by actuality, what actually is. — 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? — aletheist
Where on earth have I ever suggested that ideas are not real? — aletheist
... 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
Yes, but I claim just as insistently that numbers are nevertheless real, because ...You insistently claim that numbers have no actual existence. — 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.I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. — 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.And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed. — Metaphysician Undercover
There are two defined start points, 0 and 1, with an infinity of points between them. No end point. — Metaphysician Undercover
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
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
If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? — Metaphysician Undercover
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
What you're not respecting, is that for Aristotle ideas are part of reality. — Metaphysician Undercover
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
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
, 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
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
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
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
Word game. By "reality" I meant being actual. You've already said you don't think this is possible, — 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. — 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. — Metaphysician Undercover
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
So, what kind of infinite thing (infinity) do you think you could observe? — Metaphysician Undercover
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
We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects — Metaphysician Undercover
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.Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible. — 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.We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged
Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? — aletheist
I didn't assert a contradiction. — MindForged
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
We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged
If you think otherwise, show where measurement appears in the formalism of common geometries. — MindForged
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
this reflects conflation of the real with the actual — aletheist
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