Archimedes: The Method

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Archimedes: The Method

Post by Singular Quartet »

Alright, I don't normally post on this forum, but I figured this might go over 'well'

Archimedes, that greek guy who invented Geometery about 2300 years ago (3rd centruy BC) wrote a book called The Method, which, not only explained just about everything he came up with, but also explained how he got that, and what he was thinking. Now, this highly important math book, containing not only basic and advanced proofs of geometry, but is also being determined to contain calculus and other intresting stuff (advanced definitions of infinity) was only recently rediscovered.

The history of this book as it passed thorugh the ages is here on the PBS website (Hey, the show I learned this off of was NOVA. What do you expect?) where it survived the fourth crusade, World War 1, World War 2, and some random imbecile who decieded to draw over four of the pages as a forgery.

Now, present inspection of the book is revealing that Archimedes knew at least basic calculus, and had a deffinitve set of rules for Infinity.

Discussion Questions:

1: What do you think might have happened had this book not been written over for religious purposes, but instead been found by reniscsance scholars, who instead read it, and didn't need to rediscover all of that math?

2: How would you feel if your last work, your greatest work, was sitting in a library, and a custodian/librarian came along, looked at it, noted they were running out of shelf space, and then decieded to throw it out?

3: What was the Holy Smoot Tariff? Where did that come from?

I apologize if somebody has already posted about this.
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Post by NapoleonGH »

i was pretty sure euclid invented geometry
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Post by Master of Ossus »

1. We can only speculate. Honestly, I think it only furthers the ability of Western countries to conquer other nations in front of them, and accelerates the future, but it's virtually impossible as to speculate as to how this would have affected the future.

2. That sucks, but if my work wasn't deemed shelf-worthy I guess I wouldn't have anyone to blame but myself. In this case, however, it strikes me that the librarian obviously didn't understand the work that he was throwing out, which implies he hadn't read it, which further indicates he's not a good librarian. In that case, I would be particularly angry about the situation, but I would also assume that some people had read the work, and since this is a mathematical work that it would be re-discovered in the future. If it were a work of literature, I could only hope that it would have already reached an audience that would be affected by it, further down the line.

3. I assume this was the Harley Smoot Tariff? This was one of the most disastrous tariffs in history, and involved the US imposing MASSIVE import tariffs designed to protect American businesses during the early days of the Great Depression (about 1930, IIRC). Of course, the problem was only exacerbated by increasing costs and (more significantly) when foreign governments levied their own tariffs on American goods. Most economists agree that this was the crappiest tariff in the history of the Earth, and use it as an example of what NOT to do in a situation with a stagnating economy.
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Re: Archimedes: The Method

Post by Symmetry »

Singular Quartet wrote:Alright, I don't normally post on this forum, but I figured this might go over 'well'

Archimedes, that greek guy who invented Geometery about 2300 years ago (3rd centruy BC) wrote a book called The Method, which, not only explained just about everything he came up with, but also explained how he got that, and what he was thinking. Now, this highly important math book, containing not only basic and advanced proofs of geometry, but is also being determined to contain calculus and other intresting stuff (advanced definitions of infinity) was only recently rediscovered.

The history of this book as it passed thorugh the ages is here on the PBS website (Hey, the show I learned this off of was NOVA. What do you expect?) where it survived the fourth crusade, World War 1, World War 2, and some random imbecile who decieded to draw over four of the pages as a forgery.

Now, present inspection of the book is revealing that Archimedes knew at least basic calculus, and had a deffinitve set of rules for Infinity.

Discussion Questions:

1: What do you think might have happened had this book not been written over for religious purposes, but instead been found by reniscsance scholars, who instead read it, and didn't need to rediscover all of that math?
I already knew about this, and it still makes me cry.
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Post by Robert Treder »

Hawley-Smoot. Jeeze, people never get that one right.
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Post by Kuroneko »

NapoleonGH wrote:i was pretty sure euclid invented geometry
Oh, no. Archimedes came after Euclid [4th century BCE], who in turn came after Pythagoras [5th century BCE]. He didn't even invent the axiomatic method for geometry, as Thales [teacher of Pythagoras] is credited for that. Everything always comes back to those Milesians, the clever buggers... well, or Ionians in general.

Still, Euclid's "Elements" was much more comprehensive than anything else that came before it, so it was a major milestone.
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Post by Kuroneko »

Master of Ossus wrote:2. That sucks, but if my work wasn't deemed shelf-worthy I guess I wouldn't have anyone to blame but myself. In this case, however, it strikes me that the librarian obviously didn't understand the work that he was throwing out, which implies he hadn't read it, which further indicates he's not a good librarian. In that case, I would be particularly angry about the situation, but I would also assume that some people had read the work, and since this is a mathematical work that it would be re-discovered in the future.
Yes, but then assume that you managed to produce (1) the greatest mathematical work for nearly two millenia afterwards, and (2) this librarian also has the last copy.
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Post by Strate_Egg »

Actually, the Greeks did not invent geometry, it was used hundredes of years before they existed. SUmerians and Egyptians used it for architecture and agriculture. However, the Greeks of the classical and Hellenistic periods were excellent modifiers of the geometric ideas, especially Euclid and Archimedes.
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Post by Strate_Egg »

I still cant spell
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Post by Kuroneko »

Strate_Egg wrote:Actually, the Greeks did not invent geometry, it was used hundredes of years before they existed. SUmerians and Egyptians used it for architecture and agriculture. However, the Greeks of the classical and Hellenistic periods were excellent modifiers of the geometric ideas, especially Euclid and Archimedes.
That's true, but there is a difference of using various geometrical 'rules of thumb' as the Egyptians did in their architecture and more rigorous axiomatic derivation as the Greeks did. The Greeks were the first to do geometry for the sake of geometry, and that is significant. I'm not certain about the status of Sumerian geometry, however.
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Post by Strate_Egg »

Yea, iI agree to that. The Greeks didn't invent geometry, I considered it more a facelift of what it was.
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Post by Slartibartfast »

It's not a matter of inventing so much that it's a matter of discovering. It's like saying that Archimedes invented that an object immersed in liquid displaces a volume of liquid equal to the volume of the object... maybe before liquids didn't displace in that way?
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Post by Kuroneko »

Slartibartfast [size=75](emphasis mine)[/size] wrote:It's not a matter of inventing so much that it's a matter of discovering. It's like saying that Archimedes invented that an object immersed in liquid displaces a volume of liquid equal to the volume of the object... maybe before liquids didn't displace in that way?
Completely irrelevant. Mathematics is not physics.
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Post by Slartibartfast »

Kuroneko wrote:
Slartibartfast [size=75](emphasis mine)[/size] wrote:It's not a matter of inventing so much that it's a matter of discovering. It's like saying that Archimedes invented that an object immersed in liquid displaces a volume of liquid equal to the volume of the object... maybe before liquids didn't displace in that way?
Completely irrelevant. Mathematics is not physics.
You're right, I will rephrase that: if you discover that there's a constant between radius and circumference of a circle, you're not inventing PI.
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Post by Kuroneko »

Slartibartfast wrote:
Kuroneko wrote:Completely irrelevant. Mathematics is not physics.
You're right, I will rephrase that: if you discover that there's a constant between radius and circumference of a circle, you're not inventing PI.
Well, sure, you may discover things in a mathematical system you did not know before, but there is still a kind of confusion of levels. The system did not exist before people invented its rules (Geometry itself), and the concepts had no meaning before this system. It is similar to how might 'discover' an particularly good and previously unknown opening strategy in chess. Or will you tell me that chess itself existed before people, and was there waiting to be discovered by them?

But hey, I used to be a Platonist.
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Post by Strate_Egg »

Well, i guess it is pretty easily understood if you take it from a philosophical point of view, preferable Rationalism.


Rationalism of Plato helps people understand certain Innate ideas or self evident truths exist. For example, the Theory of Universals/Forms clarifies the existance of a metaphyscial/physical realm. The metaphysical realm contains ideal Forms or representations of ideas present in the universe. Math as well as other sciences would fall into this category.

The rules of Triangularity and Circularity would always apply no matter or not who discovered/invented them. People might not be aware of such things, but they still exist nontheless. Because we are able to "recognize" such things, we must have some type of innate knowledge of them otherwise we would not know for what we are looking. Good/Beauty/Justice/Maths
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Post by Kuroneko »

Strate_Egg wrote:Rationalism of Plato helps people understand certain Innate ideas or self evident truths exist. For example, the Theory of Universals/Forms clarifies the existance of a metaphyscial/physical realm. The metaphysical realm contains ideal Forms or representations of ideas present in the universe. Math as well as other sciences would fall into this category.
Right, this is the essential Platonism.
Strate_Egg wrote:The rules of Triangularity and Circularity would always apply no matter or not who discovered/invented them.
Apply to what? Those rules do not apply to anything in the "real world"! The only thing they apply to is the systems which people have thought up the rules for. Triangularity? Let's see... the sum of angles add to two right angles? That sum of two sides is greater than the third? Well, with the Euclidean metric, yes, but that's not the way the universe works--it doesn't even have an invariant metric (GTR is background-independent). What is this mystical Triangularity you're talking about?

Of course, by changing the system, I'm actually changing my definition of triangle, so arguably I'm not referencing the same thing. But then, to fix this while still maintaning the Form metaphysics, every logically consistent definition of 'triangle' (or anything, for that matter) would have to have its own Form. But then, there cannot be any objective, unique 'Triangularity' because every consistent definition would be equally valid. Such a metaphysics would be wholly useless, because for it everything that is logically consistent is real.
Strate_Egg wrote:People might not be aware of such things, but they still exist nontheless. Because we are able to "recognize" such things, we must have some type of innate knowledge of them otherwise we would not know for what we are looking. Good/Beauty/Justice/Maths
I disagree. For example, for a particular form (goodness), Plato's argument is basically thus: if it is possible for us to be mistaken about what is good for us, then there must be an objective standard of what is good. Plato admits only two possibilities: goodness is grounded in nomos (law--custom) or phusis (nature--objective reality). But this is a false dilemma; many philosophers have different approaches fall into neither of Plato's buckets; for example, Kant bases his morality in rationality.

If Plato has a better argument that is not a generalization of the above (as 'argument-from-recognition' would be), I cannot recall it.

Besides, the Forms' in-and-of-themselved ('auto kath auto') requirement makes for exceptionally ugly metaphysics. Relationalism is the way to go.
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Post by Strate_Egg »

Personally, i like plato's Forms, but no philosophy is without its downfalls. That i will agree to, yes. The idea of Goodness is one of the hardest to pointpoint, since as you mentioned, what we think is good is not necessarily good. Then again, in order to understand the concept of what "might" be good, we would have to already have an idea of what that good entails. The fact that we cannot always determine what is good or bad is because humans re imperfect creations of nature. The physical world according to plato isnt necessarily the true world of reality, but rather a pale immage of what should be.

And i am sorry, perhaps i was very unclear what i meant when i said the rules of circularity and triangularity would apply. What i stupidly meant was that these rules would always apply to the universe no matter if anyone were to discover or invent them. A triangle will always have the respected properties of a triangle. Circles will have their properties. Math conepts remain constant. 2 of somethign and 2 of something will always = four of something. WE can pretend this isnt true by reinterpreting the number system, but even with we change the name of the number 2, it will still represent () and (). It is a universal.

As well, logic from rationalism dictates that A and NOt A cannot exist at the same time. Something cannot exist yet not exist simultaneously. That is rationalism that relates to the Theory of Forms. We can only recognize this through rational thought as a product of innate knowledge. Empiricism relies on inductive reasoning which is not infallible. No matter how many times you try to oberve A, you won't be able to tell if it can both exist and not exist unless you use rational thinking.


Now, i am not advocating the beliefs of Descartes who belived the innate ideas were a doing of God's work; however, i find merit to the IDea of nature incsribing these truths into our being so that we could progress as a species. It would be like giving someone the tools to survival.

Plato did not mean that just because these self-evident truths exist, we all know about them. We are born with the ability to percieve them. I believe this is recollection theory.

I just think it makes sense . WHy do we know what is logical? WHy do we know what a triangle is, or a circle. WE cannot examine every circle. Additionally, no circle or triangle is perfect 100%. To the contrary; however, the IDEA of a triangle is perfect. This we recognize yet we never see and cannot sense.

Now, the forms do not govern everything, just the fundamental truths of the world. "Through education, we free ourselvs from the cave"

(Allegory of the cave, plato)

Just a question, but isnt Kant a Constructivist?
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Post by Strate_Egg »

"But then, to fix this while still maintaning the Form metaphysics, every logically consistent definition of 'triangle' (or anything, for that matter) would have to have its own Form. But then, there cannot be any objective, unique 'Triangularity' because every consistent definition would be equally valid."


Im sorry, i forgot to address this.


You CAN change the definition of a triangle, but it..no matter what...will always have three sides and the rules for triangles will always exist, no matter how someone wants to twist it.

What you are mentioning is something different than a form, rather a perversion. That is why some people refrain from calling it the Theory of Ideas. That would mean that because someone can manifest something ( a unicorn) it must exist somewhere, physically or in a metaphysical realm.


In short, if you change the definition of a triangle, then it ceases to be a triangle. A triangle will always remain a trianagle. You can molest the term, but the essence will stay. That is the form that remains.
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Post by Kuroneko »

Strate_Egg wrote:Personally, i like plato's Forms, but no philosophy is without its downfalls. That i will agree to, yes. The idea of Goodness is one of the hardest to pointpoint, since as you mentioned, what we think is good is not necessarily good. Then again, in order to understand the concept of what "might" be good, we would have to already have an idea of what that good entails. The fact that we cannot always determine what is good or bad is because humans re imperfect creations of nature.
All that shows is that we have certain preferences. For example, hedonism and utilitarianism (at least, the most common versions of it) are based on the premise "pleasure is preferable to pain" (with some additional premises depending on the type of utilitarianism).
Strate_Egg wrote:The physical world according to plato isnt necessarily the true world of reality, but rather a pale immage of what should be.
It would be better to say that Plato had many 'degrees of reality', in which objects of the the physical world, although quite real, are far from the most real objects.
Strate_Egg wrote:And i am sorry, perhaps i was very unclear what i meant when i said the rules of circularity and triangularity would apply. What i stupidly meant was that these rules would always apply to the universe no matter if anyone were to discover or invent them. A triangle will always have the respected properties of a triangle.
And what properties are those? One can't even say that the sides of the triangle have finite lengths, as it is possible for it to not be the case (e.g., in the hyperbolic plane). What good is 'three points and the lines between them' or somesuch? Personally, I find even that a bit suspect because in projective geometry, the distinction between point and line is completely arbitrary.
Strage_Egg wrote:"But then, to fix this while still maintaning the Form metaphysics, every logically consistent definition of 'triangle' (or anything, for that matter) would have to have its own Form. But then, there cannot be any objective, unique 'Triangularity' because every consistent definition would be equally valid."

Im sorry, i forgot to address this.

You CAN change the definition of a triangle, but it..no matter what...will always have three sides and the rules for triangles will always exist, no matter how someone wants to twist it.

What you are mentioning is something different than a form, rather a perversion. That is why some people refrain from calling it the Theory of Ideas. That would mean that because someone can manifest something ( a unicorn) it must exist somewhere, physically or in a metaphysical realm.
But my point is that despite Plato not intending to have things like "hair, mud, dirt" to have forms, this is precisely what his theory degenerates to.
Strate_Egg wrote:Circles will have their properties. Math conepts remain constant. 2 of somethign and 2 of something will always = four of something. WE can pretend this isnt true by reinterpreting the number system, but even with we change the name of the number 2, it will still represent () and (). It is a universal.
Or we can realize that the number system is simply a tool to model the world, not a pre-existing entity. It's a rather powerful tool, but like any other tool, it has a rather limited range of applicability.
Strate_Egg wrote:As well, logic from rationalism dictates that A and NOt A cannot exist at the same time. Something cannot exist yet not exist simultaneously. That is rationalism that relates to the Theory of Forms. We can only recognize this through rational thought as a product of innate knowledge. Empiricism relies on inductive reasoning which is not infallible. No matter how many times you try to oberve A, you won't be able to tell if it can both exist and not exist unless you use rational thinking.
Which logical system is Real McCoy, then?
Strate_Egg wrote:Now, i am not advocating the beliefs of Descartes who belived the innate ideas were a doing of God's work; however, i find merit to the IDea of nature incsribing these truths into our being so that we could progress as a species. It would be like giving someone the tools to survival.
Alternatively, those 'truths' are simply an epiphenomenon formed by Life's adaptation to nature. This view I find more plausible, since logic is learned as much as everything else. Rationality is a consequence of organization, an evolutionary process in every sense of the word.

Imagine a a car that tops out at 144mph. This fact is dictated by its design--a phlethora of variables, but nevertheless is a real fact determined by them. However, treating it as something separate (or even separable) from the car itself is pure folly. It is the same thing with rationality--it is a consequence of how humans are structured, but the jump that it has a separate reality (as Plato claims) is wholly unjustified.
Strate_Egg wrote:Plato did not mean that just because these self-evident truths exist, we all know about them. We are born with the ability to percieve them. I believe this is recollection theory.
According to Plato, everything that can be known is already known by the soul, and is acquired prenatally. But Plato seems a bit schizophrenic about the precise mechanics--Meno is incompatible with Republic, and some other works (e.g., Phaedo) present the Forms theory as religious revelation is disguise rather than any rational attempt at metaphysics.
Strate_Egg wrote:I just think it makes sense . WHy do we know what is logical? WHy do we know what a triangle is, or a circle. WE cannot examine every circle. Additionally, no circle or triangle is perfect 100%. To the contrary; however, the IDEA of a triangle is perfect. This we recognize yet we never see and cannot sense.
We know what is they are because we have defined them. For example, a circle is the set of all points that are some constant distance (radius) away from a given point (center). If my plane has a taxicab metric [f((x1,y1),(x2,y2)) = |x1-x2|+|y1-y2|] instead of Euclidean one [g((x1,y1),(x2,y2)) = sqrt((x1-x2)²+(y1-y2)²)], the problem of 'squaring the circle' is trivial because all circles are squares. (Interestingly, the converse is not true.) Before you say I've 'molested' the definition of circle, let me reiterate my point: given that the universe has no background metric, whether Euclidean or taxicab or any other, what would make you prefer one over the other? What makes one a Form and the other a perversion?
Strate_Egg wrote:Just a question, but isnt Kant a Constructivist?
Yes (though how much depends on one's interpretation of Kant), and that means he contradicted Plato on several levels. The most obvious one is that Plato's recollection theory requires all knowledge to be innate--the soul "knows the world as it really is," so to speak. For Kant, this is nonsense; there is no knowledge without the perception of the subject, so that there is no knowledge of the noumenal at all, only the phenomenal.

Not that I necessarily subscribe to Kant; the previous point was only that Kant's idea of Goodness, although objective (in the modern sense of the word), has no meaning without moral actors. For Plato, Goodness is something that exists regardless of whether being capable of moral action do. Plato 'proves' this by constructing a false dillema--the underlying assumption is that the only kind of objectivism is his kind of objectivism.

Mind you, I don't think he was into mathematical constructivism (or even considered the position), which is a wholly different kettle of fish.
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Post by Strate_Egg »

"Or we can realize that the number system is simply a tool to model the world, not a pre-existing entity. It's a rather powerful tool, but like any other tool, it has a rather limited range of applicability. "


The number system is a tool to represent the world, but rationally, the "examined" object has to exist somewhere somehow in the first place to be measured no matter whether or not you measure it at all." If that is not believed, then it delves into the realm of Bishop Berkely and his "mind dependant" view of represation of reality.

You are right, our definition of a triangle or circle may be imperfect. That is exactly the case. Our definition of something is part of the physical realm made to immitate that of the metaphysical.


"According to Plato, everything that can be known is already known by the soul,"

NOt everything, just the fundamental truths of the universe. Everything would go against the definition of Rationalism.


"Before you say I've 'molested' the definition of circle, let me reiterate my point: given that the universe has no background metric, whether Euclidean or taxicab or any other, what would make you prefer one over the other? What makes one a Form and the other a perversion? "

Now that is a very good point. The reason i prefer one over the other, is rooted in the structure of Rationalism. Rationalism provides a good explanation of reality because it uses both reason/logic, as well as empiricism. Rationalism is not limited only to reason, but to a combination of perfect reason and sensory data. Reason is just the "most perfect" source of knowledge. The properties of a circle or a triangle can be seen. We through experience can recognize "justice" even though it technically is never in perfect existance. To define a fundamental idea, it has to exist in the first place, otherwise you would be defining nothing. Now, would you advocate that the universe does not exist, or that trangles that conform to our visions of reality are false?
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Post by Kuroneko »

Apologies for my belated reply.
Strate_Egg wrote:"Or we can realize that the number system is simply a tool to model the world, not a pre-existing entity. It's a rather powerful tool, but like any other tool, it has a rather limited range of applicability. "

The number system is a tool to represent the world, but rationally, the "examined" object has to exist somewhere somehow in the first place to be measured no matter whether or not you measure it at all." If that is not believed, then it delves into the realm of Bishop Berkely and his "mind dependant" view of represation of reality.
I assume that by "examining the objects", you mean the numbers themselves.

Mathematics is not about objects. It's about structure. The "objects" of mathematical study consist entirely of their extrinsic properties. They have no intrinsic ones, so they do not refer to any specific objects. For example, consider how the Peano axioms "define" the naturals {0,1,2,3,...}. If they had independent existence, the set of odd naturals {1,3,5,...} would serve as a perfectly good model for the Peano axioms, in which the next odd number is the successor relation, and 1 is the "structural zero" in the sense of being the smallest odd natural--falling right in line with the axioms. Those two models are isomorphic. It would make not one whit of a difference to the set of propositions derived from the Peano axioms whether or not they referred to the "real" naturals or the alternative model. In fact, one can even go a step further by treating the system entirely by typographical means: if you have a number x, you can append ' to it, etc.

In short, mathematics is at best hypothetical: if one has objects satisfying a certain structure, then such-and-such follows. If not, disbelief is suspended unless there is a genuine logical contradiction. At worst, it is a game of manipulating symbols according to certain rules that have absolutely no inherent meaning. I don't see either of those positions as problematic. In fact, I see them as a strength for mathematics, since then one is free to pick any interpretation which suits one's goals and/or intuitions, whether it corresponds to anything "real" or is purely imagined, so long as it has the same structure. It is not necessary for it to mirror the real world, although that is frequently a motivation for its development.
Strate_Egg wrote:You are right, our definition of a triangle or circle may be imperfect. That is exactly the case. Our definition of something is part of the physical realm made to immitate that of the metaphysical.
But how are "real" definitions even possible? Consider the property "Px: x is prime". It can be defined as "(AyEz)(y*z=x -> (y=x v z=x))", after a suitable definition of multiplication. But how much is that a statement about x or its 'primeness' and how much is it a statement about the overall structure of numbers (it is, after all, preceded by "for all y, exists z")? The point? Only that any definitions require a background structure.

Consider how the forms would provide this structure. To prevent circularity and infinite regress of definitions, there must be primitive, undefined forms. But, if so, the whole thing would rest on things that we cannot, even in principle, have knowledge of, barring divine revelation.

One could turn to some kind of holism to solve this, but then this Platonism becomes something very, very far from Plato. Not that that's a problem. I'm just not sure what your views are.
Strate_Egg wrote:"According to Plato, everything that can be known is already known by the soul,"

NOt everything, just the fundamental truths of the universe. Everything would go against the definition of Rationalism.
Alright, but then in this case Rationalism contradicts Plato. This is straight from the Meno. One could argue the Republic implicitly contradicts this, but I don't agree.

Well, from your other posts, I gather Rationalism already contradicts Plato in the adherence to a strict principle of non-contradiction (hence my comment about logical systems). In regards to the world as perceived by the senses, Plato was a lot like Heraclitus, in that that world is full of genuine contradictions, but unlike Heraclitus (who thought that these contradictions are what makes a thing "real"), for Plato this is one of the reasons why the perceivable world is inferior to the forms, but nevertheless has a kind of reality in itself.

Let's forget the specific Platonism of Plato himself, then.
Strate_Egg wrote:"Before you say I've 'molested' the definition of circle, let me reiterate my point: given that the universe has no background metric, whether Euclidean or taxicab or any other, what would make you prefer one over the other? What makes one a Form and the other a perversion? "

Now that is a very good point. The reason i prefer one over the other, is rooted in the structure of Rationalism. Rationalism provides a good explanation of reality because it uses both reason/logic, as well as empiricism. Rationalism is not limited only to reason, but to a combination of perfect reason and sensory data. Reason is just the "most perfect" source of knowledge. The properties of a circle or a triangle can be seen. We through experience can recognize "justice" even though it technically is never in perfect existance.
Most of such concepts have an opposite (goodness<-->wickedness, beauty<-->ugliness, etc.), such that one can be defined in terms of the other.

If both are forms, then construct form "beaugly:halfway between ugly and beautiful". Surely compounding is allowed, since otherwise forms like Triangle and Circle, which are definable from 'point'/'line'/etc. (as basic concepts as one can get, and so should have their respective forms--since these forms are absolute, there would be no ambiguity due to background dependence), would not be forms. This is, of course, a silly position that is only a hairbreadth away from 'everything conceivable is real', if not outright equivalent to it.

If only one of them is a form, perhaps the problem goes away. But then, which one is metaphysically prior? Clearly one of them is excess baggage, but there is no reason to pick one over the other besides personal preference.
Strate_Egg wrote:To define a fundamental idea, it has to exist in the first place, otherwise you would be defining nothing. Now, would you advocate that the universe does not exist, or that trangles that conform to our visions of reality are false?
No, I would not. I contend that the universe does exist objectively, but that there are no objective categories (forms). Our perception of the universe occasionally filters out certain patterns which have similar structure that we have named 'triangle' for pragmatic reasons (it is useful to refer to such things) or personal interest. Certain other patterns also have similar structure, but we have no real need to refer to them, so we did not classify them into named categories. I still don't see any reason to 'formify' the triangle but not, say, a suitable generalization of a random closed figure I just drew (which I'll name 'fooangle'), other than pragmatic ones--all that shows is that we prefer to think in terms of some concepts than others. To jump from those preferences to the conclusion that the Triangularity is 'objectively real' but Fooangularity isn't is nothing short of intellectual bigotry. You claim that there is a rational way to sort them out, but I still don't see what that would be.

There is no problem if the classes of Triangles and Fooangles exist only as mental categories, not objective (mind-independent) ones.
"The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set." -- Wesley Salmon
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Kuroneko
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Kuroneko wrote:Mathematics is not about objects. It's about structure. The "objects" of mathematical study consist entirely of their extrinsic properties. They have no intrinsic ones, so they do not refer to any specific objects. ...
Hum... the reason given (italics) is actually erroneous. It is possible to get individualization of objects out of a purely non-monadic relations--in other words, extrinsic properties. However, this applies only to rather peculiarly structured relations, which the number system doesn't have. But the clincher is that this is only true under the assumption that the individuals induced by such a relation are the only things that exist in the domain, i.e., covers all the forms. This is obviously untrue in mathematics, so my previous points regarding mathematics remain unchanged. (Actually, I'm not even certain this relation form-inducement could be done for infinitely many individuals. Though I suspect that is the case, I'm only certain that is possible on finite domains.)

Applying this view to the form theory would mean that (1) the structural relation is what is the most metaphysically basic, and the forms are simply an epiphenomenon induced by it, and (2) is essentially another flavor of holism. How does your Rationalism square with that?
"The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set." -- Wesley Salmon
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Now that i think about it more, the only methodology that seems logical, concise, and strong seems Skepticism. In the end, i have to support that. It refutes most of the arguments proposed by Rationalism and Empiricism yet remains on a firm foundation according to the defintion of Knowledge.


I think its methods are safe, even thought they may seem a bit improbably. This improbability; however, is no more ludicrous than the idea of a separate physical/metaphysical realm or subjective idealism of Berkely. At least the skeptics dont try to prove sweeping assumptions based on senses, experience, or just "rational" thought.
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Strate_Egg wrote:Now that i think about it more, the only methodology that seems logical, concise, and strong seems Skepticism. In the end, i have to support that. It refutes most of the arguments proposed by Rationalism and Empiricism yet remains on a firm foundation according to the defintion of Knowledge.
Well, yes, skepticism is very logical, but in the end you must make a choice between Pure Logic and Rationality. Skepticism shows that there is no absolute knowledge. Or even perfectly reasonable belief (true or not). The rational thing to do would be something along the lines of the Humean way out--yes, there are unfounded assumptions (the very basis of Skepticism is in showing that there are always unfounded assumptions), but they have been proven to work rather consistently.
Strate_Egg wrote:I think its methods are safe, even thought they may seem a bit improbably. This improbability; however, is no more ludicrous than the idea of a separate physical/metaphysical realm or subjective idealism of Berkely. At least the skeptics dont try to prove sweeping assumptions based on senses, experience, or just "rational" thought.
I'm not certain what you mean by the 'methods' of skepticism. As far as I am aware of, Skepticism is simply an epistemic position. Please clarify.
"The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set." -- Wesley Salmon
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