I suppose you are viewing "intelligible objects" from a god-like Rationalist perspective -- from outside the world system. As far as God is concerned, everything in the world is real, and objective. But. from the human point-of-view, we depend on physical senses for most of our knowledge of reality. So, what Epistemologists refer to as a priori knowledge is literally non-sense. We obtain such god-like knowledge via reasoning from specific sensory data to generalized concepts -- which are not real things, but artificial (synthetic) propositions about holistic collections of things & logical relationships. Hence, we can only communicate those intangible ideas in terms of metaphors analogous to physical things.I parse the entire subject of the reality of ideas differently. My view is that proper 'intelligible objects' such as natural numbers, scientific principles, and the like, are real, but they're not existent things - they don't exist in the same way that regular objects do. They are strictly speaking noumenal - meaning 'objects of mind', although the sense in which they are 'objects' is debatable. — Wayfarer
Yes. But the Enformationism thesis is all about the "different ways" (forms) that things can exist. Which is what makes its phenomenal & noumenal topics so hard for some, especially philosophical Realists & Logical Positivists, to conceive. For them, you are either a truth-seeking Realist, or a fantasy-seeking Idealist. Hence, my complementary notion of BothAnd does not compute. :meh:Where that presents difficulties, is that there is no provision in most people's minds for things to exist in different ways - in other words, things either exist, or they don't. — Wayfarer
This again is incoherent. A 2d surface is a flat plane. To give that plane any type of curvature requires a third dimension. — Metaphysician Undercover
The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space.
https://en.wikipedia.org/wiki/Gaussian_curvature
This again is incoherent. A 2d surface is a flat plane. To give that plane any type of curvature requires a third dimension. You could give a line (1D) curvature, with a second dimension, but then what you get is a circular plane. — Metaphysician Undercover
I suppose you are viewing "intelligible objects" from a god-like Rationalist perspective -- from outside the world system. As far as God is concerned, everything in the world is real, and objective. — Gnomon
I'm not a Logical Positivist, but I am aware that most people apply the term "Real" only to what they can see & touch. Any other forms of knowledge are either Un-real or Ideal or spiritual or "ghostly", and consequently their "existence" is debatable. — Gnomon
artificial (synthetic) propositions — Gnomon
source.synthesis: integration of two opposing representations into one new representation, with a view towards constructing a new level of the object’s reality. Philosophy as Critique employs synthesis more than analysis. On the operation of synthesis in the first Critique, see imagination. (Cf. analysis.)
synthetic: a statement or item of knowledge which is known to be true because of its connection with some intuition. (Cf. analytic.)
The topic I’m still very interested in studying in greater detail is the significance of Kant’s ‘synthetic a priori’ and the application of all of these ideas to the subject of metaphysics. — Wayfarer
You are disputing about the most significant step forward in modern geometrical thought. — apokrisis
It's not a surprise that the mathematics we choose to talk about the world happens to fit the world, — Banno
Is your appeal to authority supposed to impress me? — Metaphysician Undercover
Can you justify your claim that space is the type of thing which can be both curved and not curved at the same time? Will you resolve this contradiction? — Metaphysician Undercover
A priori truths are an exception because they’re true by definition - the textbook example being that you can say of a bachelor that he’s an unmarried man. Even though it’s a trite example, the principle has broad scope, including (Hume would argue) mathematics and all those things we can know a priori, that is, on the basis of logic not experience. — Wayfarer
What contradiction? Even in ordinary language, flat and curved would be a pair of dichotomously opposed limits - two extremes of the one spectrum. Something would be flat to the degree it wasn't curved, and curved to the degree it wasn't flat. — apokrisis
The question for maths is how to go about measuring the relative curvature of a smooth manifold once you have got past the naive Euclidean view that space is some kind of absolutely flat backdrop.
You might rant and rave in defence of this antique view. But geometry has just got on with developing the means for modelling spaces where perfect flatness only means an extreme constraint on any intrinsic curvature.
It would help to learn more about this subject before mouthing off further. For this purpose, I would suggest Wildberger's lectures on hyperbolic geometry.
The pertinent bit is how he shows that the Euclidean yardsticks developed for measuring spaces without curvature - distance and angle - must be replaced by the new dichotomy of quadrance and spread when dealing with hyperbolic "flatness".
So there is nothing arbitrary going on as it is all motivate by the rigorousness of dialectical argument.
And Appollonius had already worked out the basics for this approach back in 200 BC.
So even if your knowledge of maths is still rooted in distant antiquity, you ought to know better.
See Wildberger's lecture series - https://youtu.be/EvP8VtyhzXs — apokrisis
A Wittgensteinian answer to this question would that there is no such thing as physical causation as is generally understood in modern science, but that physical causation is an a priori intuition, which is useful for hypotheses, but which tells us nothing about the world in-itself or its meaning.
A flat thing has zero curvature. And anything which is not flat has some degree of curvature. — Metaphysician Undercover
So all degrees are degrees of curvature, and flatness has no degrees, flat is zero degrees of curvature — Metaphysician Undercover
I think it’s more likely that either people don’t understand it, or don’t want to engage with Kant’s work. — Wayfarer
That’s a lame attitude, it seeks to explain away the ability of reason to make genuine discoveries - to uncover, disclose, intuit, things about the world which were never previously known and could not be known by any other means. It’s a kind of covert neo-Darwinism which reduces everything about humans to an adaptation, it’s what I mean about modern philosophy being basically irrational (or sub-rational). — Wayfarer
People do maths, not god. — Banno
I get it and within this hints of idealism. — Tom Storm
So, you don’t like him? — Wayfarer
So, you don’t like him? — Wayfarer
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