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JM

Apparently you are not aware of intrinsic geometry. It is not possible to decide a priori which hypothesis concerning parallels is the right one. Einstein turned geometry from an axiomatic science into an empirical one. We know that Euclid was wrong about parallels, the angle sum of triangles, and many other things. The ancients gave a good first approximation, but as with so many things, their ideas have been overturned given greater experience and better measurements.

When you say that "curved"straight lines (geodesics?) are really curved, by what notion? We do not need to assume extra dimensions into which they curve. They are straight from the perspective of observer's within their space. Maybe it's the parallel postulate that is empirically false.

Also, what about the continuum hypothesis? Are there infinite cardinalities between the countable infinity and the first uncountable infinity? If so, how many? Are there an infinity of infinities between these two infinities? If so, which cardinal infinity? Or, is there something fundamentally wrong with a theory that splits infinities? If so, then surely math is not a priori but rather contains man-made aspects.

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Eiuol, I think what you said pretty close sums up my position.  I would make a few minor modifications. 

  • Rather than saying "the justification does not always require experience," I would say that "sense experience alone cannot justify many of the conclusions of math and science."
  • About Godel, I would put it something like this: "Godel proved that mathematics is not merely axiomatic--and thus for us to really know mathematics, we must have knowledge of some mathematical objects that is more than merely human-made definitions." 
  • To your statement about observations, I would put it this way: "observations can falsify, but can never verify the universal statements at the core of every scientific theory."
  • About thought experiments: I think they do often use logic, and they also may use our temporal-spatial intuitions. 

 

I think the "a priori categories" that you mentioned are real.  I can define them and say what I think they are.  I would define them as that according to which abstraction must take place, and I think they include concepts like place, time, kind, and cause.  By "categories," I mean fundamental concepts at the back of all other concepts, which cannot be defined using other concepts, but only with synonyms.  

 

When I say "I don't know what they are," I mean that I don't know ontologically what they really actually are.  (Are they like Plato's forms?--ideas in the mind of God?--features of all possible worlds?--clusters of neurons in our brains?)  I don't think we need to solve that problem in order to believe in the existence of a priori categories.  (In the same way, we don't really know what numbers--for example--are, and they are I think subject to the same possibilities mentioned here….)

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From John #21

"There are many things that we know by association without knowing what exactly they are--including matter, people, beauty, and just about everything in the world!"

 

From John #27

"When I say "I don't know what they are," I mean that I don't know ontologically what they really actually are."

 

From John #32

"But we can argue for the existence of something (and I think even have certain knowledge of the existence of that thing) without being able to say exactly what that thing is in its essence.  I think we know that we exist, but the idea of the self and its essence is wrapped in mystery and even paradox.  Matter is another familiar example.  We know it exists by constant experience, but even scientists cannot say what--at bottom--it really is, and all their explanations (to date) end in more mystery and paradox."

 

John,

Objectivism holds that Essence is not ontological, but rather epistemological and contextual.

 

In some context, a human is an animal with opposable thumbs.  In other contexts, man is the animal capable of abstract thought and language.  In other contexts, man is a bi-pedal animal.  In other contexts, man is a mammal (as opposed to a reptile).  In others, he's a living organism (as opposed to a rock).

 

There is no one, single characteristic which is more "essential" than any other-  except within a given context.  There is no "at bottom" definition of what ANY concept is.  This is true for oak trees, stop lights, language, math, numbers, sodium, electricity, NFL Football, etc.

 

Math and numbers are not some "ontological essence" which are floating independent of man's mind and perceptual mechanisms (eyes, ears, etc.).

 

There is an interesting concept know as subitizing which I belive lies at the heart of all mathematics - an it is not a feature unique to just humans.

Edited by New Buddha
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Aleph, I'm not an expert in advanced mathematics, but my understanding of the situation has been as follows.  You can tell me where / if I am misunderstanding and guide me as to how I can come to understand. 

 

First, my understanding of non-Euclidian geometry is that it is essentially geometry in which the parallel postulate does not hold (and those about triangles adding up to 180 degree, etc.).  Secondly, that it is impossible to decide from the axioms of mathematics, what type of geometry (Euclidian or not) describes the universe.  Thirdly, that in the theory of General Relativity, space is curved and so non-Euclidian geometry is used to describe it.  And this has been tested empirically and is not doubted by scientists. 

 

But I have never seen the ideas discussed without reference to the way that objects which appear 2 dimensional from one point-of-view are actually 3-dimensional, such as this: http://en.wikipedia.org/wiki/File:Triangles_(spherical_geometry).jpg

 

In this example, it is clear to me that the object in question is not a triangle, so the normal "laws of triangles" do not apply to it.  It LOOKS like a triangle if you are directly overhead, but in fact, since the sides are not linear, it is plainly not a triangle.  Thus to say that this shows that Euclid's axioms do not hold seems to me to be cheating.  They do hold--just not when you are describing something in 3 dimensions at one time (when you are showing how and why the axioms do not hold), and in 2 dimensions at another time (when you are applying that conclusion as some sort of sweeping undermining of all of mathematics).  That is equivocation, and I don't accept it. 

 

You might say, "but the universe is non-Euclidian, and therefore Euclid's axioms really don't hold!"  But I think they do still hold, just not in either hypothetical or real "curved space." 

 

Or perhaps I am misunderstanding something, and if so, what?

 

I'm not sure what to say about the continuum hypothesis.  I'm afraid I don't have anything mind-blowing to say about it.  It seems to me to be self-evidently false.  Obviously that can't be proven.  I fail to see the relevance to this discussion though, except that if it were in fact self-evidently false and unprovable, that would lend further support to the idea that mathematics is not merely axiomatic.  But I'm sure I'm over my head in even commenting on it, which is why I've tried to avoid doing so. 

 

New Buddha, how does objectivism square itself with Godel's proof?  According to objectivism, is there such a thing as "the way things are?"  

Edited by John Molineux
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JM,

 

The spherical representation of a triangle having angle sum greater than a straight angle is merely a representation having an imbedding in 3-D space. Mathematically, such representations as imbeddings are unnecessary. What is more, one should ask what geometry could be discovered from the perspective of someone confined to live within the space in question. After all, it is not possible to remove ourselves from our own space. When you refer to those paths in the image that you shared as curved, you are using an extrinsic notion of "curved". They appear curved when looking from outside the space in question. What if it were not possible to look from "outside" the space? From the intrinsic perspective, they are not curved. Here is another example. Take a piece of paper and draw an X from corner to corner. These lines are not curved as the plane lies flat on the table. Then make a cylinder out of the paper by joining the top and bottom corners. From the perspective of someone confined to live within such a space, locally the parallel postulate would be true and triangles would have angle sums that equal a straight angle but intersecting lines would intersect more than once as you can see from your construction. How would someone who is contrained to live within such a space know that their space is not quite Euclidean on large enough scales?

 

How do we known the intrinsic geometry of the universe within which we live? Any imbedding of the universe within a larger space is unphysical and of no relevance to us. We can only discover the geometry of our space through intrinsic means. We know of gravitational lensing and hence of light rays behaving like those straight lines of the cylinder--intersecting more than once. Gravitational paths are unaccelerated paths and hence act like geodesics of differential geometry. Geodesics are paths that are as straight as possible on spaces that may be bent. Is our universe bent? We can only answer this question empirically. Empirically, the answer is that a curved model for spacetime gives the most accurate predications for what is actually observed. These predictions have been tested recently by Gravity Probe B to within milli-arc second accuracy.

 

Concerning the Continuum Hypothesis, since that is the premier application of Godel's Incompleteness Theorem, I thought that you should certainly be interested in it. Whether you accept it or not is truly up to you. Standard math accepts it but many set theorists do not. It is optional and hence not a priori. The same is true of the rest of mathematics. There are forms of arithmetic where 2+2=0. These forms of arithmetic have found many applications. Surely you have misinterpreted the philosophical implications of Godel's Incompleteness Theorem.

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John asked : 
 

how does objectivism square itself with Godel's proof?  According to objectivism, is there such a thing as "the way things are?"

 
Unequivocally yes, there is a such thing as "the way things are" for Oism. Its called the primacy of existence principle which is an application of the law of identity..   Oism does not hold to anything like a Kantian neumenal world. Rather Oism holds that existence-identity is "at bottom" of the entire hierarchy of knowledge and there is no context where this absolute-universal concept is not implicit.
 
Concerning the Oist view of essence, and the context of context, Ms. Rand said:
 

An objective definition, valid for all men, is one that designates the essential distinguishing characteristic(s) and genus of the existents subsumed under a given concept—according to all the relevant knowledge available at that stage of mankind's development.

Introduction to Objectivist Epistemology

 

This must not be ignored in the total of what context is to Oism.... Given the totality of what is known man is essentially and objectively the rational animal. That is, this aspect of man which serves to distinguish men from all other existents has a metaphysical basis upon which the subject abstracts it from.

 

I'm working on a detailed post in response to you John. There are a host of tangled concepts in your comments which I will address today.

Edited by Plasmatic
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Eiuol, I think what you said pretty close sums up my position.  I would make a few minor modifications. 

  • Rather than saying "the justification does not always require experience," I would say that "sense experience alone cannot justify many of the conclusions of math and science."

That doesn't mean that the justification can be "rationality" alone, which would be a priori. I haven't argued that experience, on its own, justifies anything. If you're saying that experience plus other processes is required, I agree. But that's not a priori - justification still needs experience there. I wonder if you take a priori just as what you can do without presently experiencing what you're justifying? All along, I've only been saying experience was required at some point for those justifications to adhere to reality.

Let me go back to why I kept insisting on talking about knowledge formation. To form knowledge, rather than a sense of truthiness and "feeling" something is true, we have to justify it as knowledge. If you can form knowledge without justification, then why do you call it knowledge? The knowledge would be neither right not wrong when there is no justification. Or, it comes pre-justified as knowledge, but you already denied that by denying inborn knowledge. So when you separate formation and justification, there's a problem - I would say the formation is justification. You might say you're focusing on justification as one step, so formation is still experience-oriented. Foundational to experience maybe, as without justification, who knows if knowledge is possible? You still need to explain how a complete justification needs no experience, though.

Even taken as one step in a whole, a justification needs to correspond to reality or else I might as well let my imagination run wild. What possible correspondence with reality is possible without involving experiences? No, experiences alone don't do the trick, but we need them as a means to compare our beliefs to reality. To be sure, experience is a different "language" than formal expressions, so we can't treat experiences as direct-to-proof or as formal expressions. There is no direct translation. However, it is possible to describe perception and experience. If thought experiments and intuitions justify anything, it's when they correspond to something I've perceived. Now, I may be wrong about my descriptions or what I say about my perception, but my justification is still using my perception.

Once I justify, say, the number 2, I can then use 2 for further knowledge, all without re-justifying 2. Anything that follows from 2 is, by extension, necessarily justified with perception in part as far as everything related to 2 is concerned.

"By "categories," I mean fundamental concepts at the back of all other concepts, which cannot be defined using other concepts, but only with synonyms.  "

If you mean "fundamental" as developmentally fundamental for learning, why do you suppose there are a priori categories? If you mean "fundamental" as the "true essence" of reality, I simply reject that notion entirely for the reasons New Buddha gave.

"(In the same way, we don't really know what numbers--for example--are, and they are I think subject to the same possibilities mentioned here….)"

You walked into this! :)

http://existentialcomics.com/comic/36

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Aleph, I think we have an example of geometry being discovered from the perspective of someone confined to live in a lesser-dimensional reality (whatever that means)--namely, this world, where the development of non-Euclidian geometry and the discovery that the universe consisted of curved space occurred was accomplished despite the limitations of our intuitions (and our inability to intuit anything beyond 3 dimensions). 

 

Even if it were not possible to see that those lines are curved, the essential fact is that they ARE curved, regardless of how they appear.  Limited perspective does not nullify truth!  Objects look bigger when they are closer to us.  Are we ready to say therefore that they ARE bigger?--knowing, as we do, that it is possible to walk up and measure them?  I've always felt that it was just bad philosophy that wanted to draw big important conclusions from the riddles created by the limitations of various perspectives (if a tree falls and no one hears its sound….). 

 

I think we can only know whether space is curved by experiment and observation working within a framework of a priori intuitions (as we have).  But its important to note that mathematics remains the same.  You can talk meaningfully of 2-dimensional math in a 10 dimensional universe, or visa versa, and in all universes the same laws apply for the given geometrical system.  The object of math is math (not the physical universe). 

 

It's not clear to me how the Continuum Hypothesis is the premier application of Godel's theorem. 

 

Also, I'm not aware of math where 2+2=0.  I would be quite surprised if it were anything more than a matter of changing the rules for the symbols or something, since I believe myself to know with as much certainty as anything that 2+2=4!

 

Plastmatic, New Buddha told me that "objectivism holds that essence is not ontological, but epistemological and contextual," and that "there is no characteristic which is more essential than others," and no "at bottom definition."  That seems to contradict the "objective definition," mentioned by Ms. Rand.  Moreover, I think Ms. Rand's definition does not seem to me to even go far enough to reach even plain everyday truth--in which a truth is a truth regardless of the particular stage of mankind's development at the moment.  As you said, "there is such a thing as 'the way things are'."  If so, the truth is that (the way things are), again, regardless of mankind's developmental progress. 

 

Eiuol, I'm not arguing that all knowledge is a priori--just that we have some a priori knowledge which combines with experience to form the conclusions of science.  Thus I fully agree that experience is required for scientific knowledge.  I find it hard to believe that anyone could deny that! 

 

In the debate between empiricism and rationalism, empiricists say that "all knowledge comes from sense experience," but rationalists do not say correspondingly "all knowledge is a priori."  Rather, rationalists acknowledge that much knowledge clearly comes from sense experience, but just not ALL knowledge--some of it is a priori. 

 

I take your point about the connection between justification and formation.  You're right in pointing out that they are essentially the same.  Insofar as something really is knowledge, the only way we can come to have it is by experiencing / thinking through the justification. 

 

About categories, by "fundamental" I mean that understanding itself rests on them, and they cannot be understood by definitions using other words.  It seems self-evident to me that there are such categories, or else understanding could never take hold.  It is  impossible to have a free-floating circular structure of concepts where A is defined by B, B by C...and so on, until Z is again defined by A---this is not only manifestly impossible, but not the case.  When you actually extrapolate out concepts by their definitions you find at the bottom certain fundamental concepts which are defined only with synonyms, and thus which are NOT (like other concepts) understood by their definition, but in some other way--namely, directly.  These are categories. 

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There are several confusions in this thread about incompleteness, mathematical logic, and set theory that I comment on here. I might not have time to address responses or disputes to this post.
 
"Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof." [John Molineux]
 
I don't wish to opine on the above as a description of the philosophy of formalism in mathematics, but I can say a few things. The philosophy of formalism in mathematics is usually centered on the ideas Hilbert and while I am not an expert on Hilbert's philosophy of formalism, I think I know enough to note that his view is often badly mischaracterized, especially on the Internet. Most saliently, the notion "mathematics  is just a game of symbols" is often ascribed to Hilbert (though not by the poster), but there does not seem to be anything written by him that justifies characterizing his view that way. While Hilbert stressed the finitary/concrete/contentual aspect of mathematics, to my knowledge, he did not claim that mathematics does not involve mind, abstraction, or conceptualization, and whatever he might have said about analyticity, I don't know that he ever wrote anything that is fairly paraphrased as "mathematics is just a matter of extrapolation form concepts whose definitions we create"). Hilbert did famously embrace Cantorian set theory with its abstract infinite sets. What Hilbert hoped to do was prove in a finitary way the consistency of the mathematics of infinite sets (in later context, we would refer to the formal theory ZFC rather than Cantor's unformalized work itself). Arguably, the second incompleteness theorem does defeat Hilbert's hope, though it would take a more confident writer than me to opine that this is conclusive.  
 
/
 
"Godel proved that you cannot prove the Continuum Hypothesis. Paul Cohen proved that you cannot disprove the Continuum Hypothesis." [aleph_1]
 
That is incorrectly switched. 
 
Let us say that the consistency of ZF is taken as a tacit condition for these relative consistency matters: 
 
Godel proved that ZF does not prove the negation of the axiom of choice (i.e., he proved the relative consistency of ZFC) and that ZFC does not prove the negation of the continuum hypothesis (CH) nor of the generalized continuum hypothesis (GCH) (i.e., he proved the relative consistency of ZFC+CH).
 
Cohen later proved that ZF does not prove the axiom of choice (i.e., he proved the relative consistency of ZF+~AC) and that ZFC does not prove CH (perforce GCH) (i.e., he proved the relative consistency of ZFC+~GCH). 
 
"You say, "Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof." It seems that Godel's Incompleteness Theorem implies quite the opposite of what you are saying. For example, Godel proved that you cannot prove the Continuum Hypothesis." [alpeh_1]
 
As I mentioned, that is not what Godel proved. And it is not well put to say that the relative consistency and independence of the continuum hypothesis is an implication of the incompleteness theorem. And I don't see how the relative consistency and independence proofs refute a refutation of formalism. By the way, speaking of the independence of the continuum hypothesis, proved by Cohen, we note that Cohen considered himself a formalist. 
 
"It follows that this hypothesis is beyond logic and represents pure rationalism since it is also beyond observation." [aleph_1]
 
I don't know why we would have to go to such elaborate independence results for this, depending on what is meant by "logic". These independence results are not discussed by mathematicians in a sense of 'logic' limited to Ayn Rand's definition of 'logic'. In the context of the mathematical logic in which such independence results reside, there are theorems of (pure) logic, such as those of the mere first order predicate calculus, and there are specifically mathematical theorems that are not theorems of pure logic (they are non-logical theorems) . Even "0+0=0" is such a mathematical theorem that is not a theorem of pure logic but rather depends on non-logical mathematical axioms. Granted, theorems such as "0+0=0" are finitary, but still we can easily point out non-finitary theorems that, without such elaborate results as the independence of the continuum hypothesis, trivially we can show not to be theorems of pure logic. Whether such theorems are "rationalism" per Objectivism, I would leave for Objectivists to say. Meanwhile, without involving such elaborate results as Godel's or Cohen's we can trivially show that the axiom of infinity itself is a non-logical axiom. Arguably, this alone defeats Russell's philosophy of logicism that wishes to prove the theorems of mathematics (say, real analysis) from purely logical axioms though, it would take a more confident writer than me to opine that this is conclusive. 
 
"Godel showed that mathematics contains theorems that cannot be proved through mathematical logic, and that mathematics cannot be used to prove its own consistency." [aleph-1]
 
By definition, in the context of the mathematical logic in which this work is done, a 'theorem' is a statement that is proven from some set of axioms. What you may be referring to is that Godel-Rosser showed that for any consistent recursive axiomatization, there are true statements about natural numbers that are not theorems from said axioms. Meanwhile, of course there are theorems of mathematics not provable by pure logic, but that is a rudimentary fact of beginning logic that does not need anything so elaborate as the incompleteness theorem. 
 
/
 
"Do you relate the self reflexive nature of Godel.s theorom to the type of tautalogical circularity of axioms?"  [Plasmatic]
 
What "self-reflexive nature of Godel's theorem"? What tautological circularity of what axioms? Of course, the purely logical axioms are validities - either tautologies of sentential logic or validities of predicate logic. In any case Godel's incompleteness theorem itself is provable from purely finitistic assumptions: Proof of the incompleteness theorem may be shown by reasoning confined to intuitionistic logic and mathematical assumptions that do not extend past those of basic arithmetic. 
 
/
 
"But such statements can never be justified by pure experience, per the problem of induction.  And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)!" [John Molineux]
 
The Godel-Rosser incompleteness theorem is (in contemporary terminology) that for any recursively axiomatized, consistent, "arithmetically-adequate" (my term for a notion that requires more technical elaboration) theory (such as Robinson arithmetic (Q) or primitive recursive arithmetic (PRA)), there are statements in the language of the theory such that neither the statement nor its negation are theorems from said axioms.  And this proof may be given from merely finitistic assumptions. And, for such a theory, there are true statements of arithmetic that are not theorems of the theory. Moreover, for such theories, there is an algorithm to produce a statement such that neither it nor its negation is a theorem. Whether this suggests about analyticity, it is not part of what was PRECISELY proved, but rather requires philosophical argument from the mathematical result itself. 
 
/
 
"Godel showed that no axiomatic system can prove itself." [Eiuol] 
 
Where does Godel show anything like that? What does it even mean to say that an "axiomatic system proves or does not prove itself"?  An axiomatic system proves theorems, but an axiomatic system itself is not something that is proved. This does not require anything from Godel. Perhaps what you have in mind is that Godel's second incompleteness theorem is that no recursive, consistent, arithmetically-adequate theory can prove its own consistency. This has nothing to do with "a system proving itself" nor does it even entail that there are not axiomatic systems that don't prove their own consistency. 
 
"Godel proved that math is -only- a matter of extrapolating concepts." [Eiuol]
 
What specific proof by Godel are you referring to? The incompleteness theorems are mathematical results. Claims such as "math is only a matter of extrapolating concepts" are philosophical. Godel did have philosophical follow-ups to his mathematical results, but I don't know what writings by Godel or in the philosophical literature you have in mind. 
 
/
 
"[...] what about the continuum hypothesis? Are there infinite cardinalities between the countable infinity and the first uncountable infinity?" [aleph_1]
 
No, of course there are not. It would be self-contradictory to say there is an uncountable cardinal less than the first uncountable cardinal. And that has nothing to do with the continuum hypothesis.
 
The continuum hypothesis is that the cardinality of the set of real numbers is the first uncountable cardinal. Whatever the case may be about that, it couldn't possibly bear on the fact that, by definition, there is no uncountable cardinal less than the first uncountable cardinal. 
 
"Concerning the Continuum Hypothesis, since that is the premier application of Godel's Incompleteness Theorem [...] Standard math accepts it [...]." [aleph_1]
 
I don't know what is meant by "the premier application". In any case, we don't need to go to the continuum hypothesis to find prominent, "substantive" and, indeed, arithmetical exemplars of the incompleteness theorem. 
 
And what "standard mathematics" accepts or depends on or even has anything to do with the continuum hypothesis? 
 
/
 
 "I'm not aware of math where 2+2=0." [John Molineux]
 
For example, modular arithmetic for, say, a clock that has four positions 0, 1, 2, 3 and "+' for moving clockwise among positions.
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John said:
 

Plasmatic, in reference to the article you cited about the analytic-synthetic distinction. The author says,
 
"definitions that are formed with reference to things in the world are called a posteriori. (They are posterior to experience.) Those that could be formed without reference to the way the world is would be a priori. (They are prior to experience.) But concepts can’t be formed without reference to the way the world is."
 
I don't accept that way of defining a priori. Rather than the distinction being about what is referenced by the concept, it is about what justifies them. While a posteriori statements are justified by sense experience alone, a priori statements are justified independent of experience. The author himself admits the following:
 
"Such mature essentialized definitions are the building blocks of the exact sciences. They make it possible to have truly universal statements, statements that allow no exception. If what you measure doesn’t come out to be the ratio of voltage to current, then what you are measuring is not resistance. Simple. If the angles don’t add up to 180°, then the figure isn’t a planar triangle, because that sum can be derived from the very definition of a planar triangle."
 
That is pretty much what I'm saying in this post. But such statements can never be justified by pure experience, per the problem of induction. And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)!

 
John, you have moved the goal post a few times on what you think the apriori is:

I should clear define what I mean by a priori. I mean something that we "just know" and which can neither be provided, nor justified, by sense experience. ------
 
you're right about how I would define a priori. Sense experience does cause it to arise, but it goes beyond what sense experience could provide. For example, when I learn about the properties of triangles, I come to know certain things about how triangle must always be--which constitutes universal knowledge (which sense experience could not provide).---
 
intuitive is a synonym for a priori (by my meaning).---
 
I would say that at least in practice experience is necessary intuition to conclude anything--
 
even if its just the experience of thinking. But, in the real world, clearly our conclusions arise out of a combination of sense experience and thought, both of which children progress in as they grow. But thought about sense experience itself can't produce the kind of knowledge that we have without something else---
 
the thinking that underlays the scientific method relies on a number of concepts which can (in my mind) never be justified or taken from sense experience, and out of which actual universal knowledge (such as that of mathematics) can be extrapolated (I'm thinking things like math, logic, cause-and-effect, etc.). That's what I mean by a priori. ----
 
I don't accept that way of defining a priori. Rather than the distinction being about what is referenced by the concept, it is about what justifies them. While a posteriori statements are justified by sense experience alone, a priori statements are justified independent of experience.[...]such statements can never be justified by pure experience per the problem of induction. And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)! ---
 
Logic is what you can use FROM 1+1=2. But I am talking about how you get there in the first place. Neither logic alone, nor experience alone, nor the two working together can get you there! ----
 
I've repeatedly accepted that experience causes a priori knowledge to arise, but that I define it as that which cannot be justified by sense experience alone.----

 
We have a mess here to unravel. My primary goal is to steer this conversation back to the context of this forum which is Objectivism.
 
Objectivism has a distinct view of what "intuitiion" is and nowhere does Ms. Rand support anything like "intuitions are a valid basis" for one to prove anything. Intuition is a species of the genus mysticism.

Mysticism is the theory that man has a means of knowledge other than sense perception or reason, such as revelation, faith, intuition, and the like. As we have seen, this theory reduces to emotionalism. It amounts to the view that men should rely for cognitive guidance not on the volitional faculty of thought, but on an automatic mental function,---- OPAR

Reason is the faculty that identifies, in conceptual terms and by the method of logic, the material provided by man's senses. "Irrationalism" is the doctrine that reason is not a valid means of knowledge, nor a proper guide to action; "mysticism" is the doctrine that man has a non-sensory, non-rational means of knowledge.----
The Objectivist—October 1969
Nazism Versus Reason

 
In the section titled Abstraction from Abstractions in the appendix to ITOE, Ms. Rand answers a question where Prof C uses the term "intuitively" and she uses scare quotes while saying:

Prof. C: I know that I see similarities not only at an elementary level but between theories and so on. But my question is: can you identify the mental steps that one takes in grasping, intuitively, more abstract similarities? For the perceptual level, I find the answer satisfactory that to look is to see, but I'd like to know the process for the more advanced stages.
   AR: The answer is the same: to look is to see—but what constitutes a "look" on the conceptual level? Your grasp of the referents of the concepts you are dealing with. All that you need to do is to look—remembering what a look means on the conceptual level: identify what you mean by your concepts not only definitionally, but by reducing them to the facts of reality, to the perceptual level. Identify what kind of things your concept refers to in reality. And by defining and concretizing that, you will see the differences and similarities as easily as on the perceptual level.
   If you want to know why, with regard to abstractions from abstractions, you can "intuitively" know which are similar and which are different, that's really a psycho-epistemological question. Your ability to see abstract similarities "intuitively" simply stems from the fact that you have automatized that knowledge. For a human being's purposes of dealing with an ever larger amount of knowledge, a great deal of automatization is required. You would never get very far if you had to constantly and consciously retrace all the steps of how you formed each concept and then compare each concept from scratch with each new one.
   What is automatized is your understanding of what you mean by those concepts. And if you go very high in the conceptual chain, you may be unable to identify instantly why a certain high-level concept seems to you to have similarities with another. You will say, "I know it intuitively." <ioe2_221>
   What do you mean by that? That the knowledge has been automatized in your mind. Then what's needed, if you want to prove it to somebody who doesn't see it, is to break down that automatization—that is, to identify exactly the defining characteristic, and even some of the lesser characteristics, if necessary, of the concept. You give an exact statement of what you are dealing with and where you see the similarity and the difference.

 
Now, The above does relate directly to the Oist view of the error in Aristotle's view of essence. Aristotle used the concept just as you have on several occasions. Instead of the grasping of similarities and differences and omitting quantitiave measurements of a group of similar existents, he thought that there was a sort of platonic universal that inhered in entities that gave man the ability to grasp essence by a passive process of intuition.

Let us note, at this point, the radical difference between Aristotle's view of concepts and the Objectivist view, particularly in regard to the issue of essential characteristics.
It is Aristotle who first formulated the principles of correct definition. It is Aristotle who identified the fact that only concretes exist. But Aristotle held that definitions refer to metaphysical essences, which exist in concretes as a special element or formative power, and he held that the process of concept-formation depends on a kind of direct intuition by which man's mind grasps these essences and forms concepts accordingly.
Aristotle regarded "essence" as metaphysical; Objectivism regards it as epistemological.[...]
 
Aristotle proceeded from a certain erroneous metaphysics. He assumed that there are such things as essences—and that's the Platonism in him. But he didn't agree with Plato's theory that essences are in a separate world. He held that essences do exist, but only in concretes. And the process of concept-formation, in his view, is the process of grasping that essence, and therefore grouping concretes in certain categories because they have that essence in common. It is the same essence, but in different concretes.
You see, he approaches the subject from that perspective. He isn't concerned with perceived similarities and differences. And since he can't explain how it is that we grasp these essences, which are not perceived by our senses, he would have to treat that grasp as a direct intuition, a form of direct awareness like percepts, but of a different order and therefore apprehending different objects.

 
 
All this discussion about babies vs adults and the apriori is colored by your framing the problem in terms of "justification" as against formation, or reference. 
 
John said:
 

I believe that it is both true that we have a priori intuitions which underlie much of our knowledge, AND that those intuitions are awoken by experience. Again, the distinction (as I define it) has nothing to do with the conditions under which knowledge comes to arise--rather, it is about its justification.-----

 
Since composing the above you have responded to Louie in acceptance that justification is a type of formation, or rather it is a retracing of the process of formation.  Now for you to apply this to what you call the apriori ....
 
We have to untangle many of your categorizations before we can go on to demonstrate how Oism validated what you called "fundamental concepts at the back of all other concepts, which cannot be defined using other concepts, but only with synonyms.".
 
That is, what we are really discussing here is what Oist call axiomatic concepts.
 
Your use of axioms as relates to mathematics and Gödel may not be commensurate to the Oist view of axioms. The issue of definition has come up but we have to differentiate the different kinds of concepts and how definition applies to the differing kinds.
 
 

Ostensive definitions are usually regarded as applicable only to conceptualized sensations. But they are applicable to axioms as well. Since axiomatic concepts are identifications of irreducible primaries, the only way to define one is by means of an ostensive definition e.g., to define "existence," one would have to sweep one's arm around and say: "I mean this."

 
 You see that Oism rejects your view that philosophic primaries are merely awoken or awaiting experience to become quickened-activated or some such. The referents of axiomatic concepts are implicit in ANY state of awareness. This can be called acontextual in the sense that there is no context where these concepts don't apply.(David Kelly uses this terminology somewhere) This is what differentiates them from other concepts that are defined by a species and genus definition in the form of other concepts. The definition preserves the generative context of differentiation the concept was abstracted from. The particular "to's" and "from's" which give the concept meaning and make possible differentiation and integration.
 
John said:
 

in the real world, clearly our conclusions arise out of a combination of sense experience and thought, both of which children progress in as they grow.  But thought about sense experience itself can't produce the kind of knowledge that we have without something else. ----
 
 If a priori was a matter of the conditions under which knowledge arises (and was inconsistent with any knowledge that arises from experience), then newborn babies would necessarily have as much a priori knowledge as adults (since the thing that separates adults from babies is experience, and if experience caused the knowledge-difference between the two and yet was inconsistent with a priori knowledge, it would follow that babies would have to have all the a priori knowledge which adults do).

 
The Oist explanation of the difference here in regards to axiomatic concepts is what is called "implicit knowledge".
 

Axiomatic concepts identify explicitly what is merely implicit in the consciousness of an infant or of an animal. (Implicit knowledge is passively held material which, to be grasped, requires a special focus and process of conscious-ness—a process which an infant learns to perform eventually, but which an animal's consciousness is unable to perform. )----
 
 
The building-block of man's knowledge is the concept of an "existent"—of something that exists, be it a thing, an attribute or an action. Since it is a concept, man cannot grasp it explicitly until he has reached the conceptual stage. But it is implicit in every percept (to perceive a thing is to perceive <ioe2_6> that it exists) and man grasps it implicitly on the perceptual level—i.e., he grasps the constituents of the concept "existent," the data which are later to be integrated by that concept. It is this implicit knowledge that permits his consciousness to develop further.
(It may be supposed that the concept "existent" is implicit even on the level of sensations—if and to the extent that a consciousness is able to discriminate on that level. A sensation is a sensation of something, as distinguished from the nothing of the preceding and succeeding moments. A sensation does not tell man what exists, but only that it exists.)
The (implicit) concept "existent" undergoes three stages of development in man's mind. The first stage is a child's awareness of objects, of things—which represents the (implicit) concept "entity." The second and closely allied stage is the awareness of specific, particular things which he can recognize and distinguish from the rest of his perceptual field—which represents the (implicit) concept "identity."
The third stage consists of grasping relationships among these entities by grasping the similarities and differences of their identities. This requires the transformation of the (implicit) concept "entity" into the (implicit) concept "unit."
When a child observes that two objects (which he will later learn to designate as "tables") resemble each other, but are different from four other objects ("chairs"), his mind is focusing on a particular attribute of the objects (their shape), then isolating them according to their differences, and integrating them as units into separate groups according to their similarities.
This is the key, the entrance to the conceptual level of man's consciousness. The ability to regard entities as units is man's distinctive method of cognition, which other living species are unable to follow.

ITOE

 
Implicit knowledge is not merely what is possible or able to be conceptualized but actually constitutes a type of awareness that upon explicit conceptualization-verbalization one realizes they were already making these differentiations on a perceptual-visual level (Dr. Peikoff mentions this in the 1976 lectures).
 
The difference between adults and children as regards axiomatic concepts is not a lack of awareness-experience of the referents of axiomatic concepts but a lack of the cognitive tools provided by conceptualization-language.
 
That is, the key thing you are missing. The void in your validation-justification of these philosophical primaries which you are taking recourse to "intuition" to fill, is the method whereby one becomes aware of the axiomatic status of these concepts. Aristotle called this reaffirmation through denial.
 

"An axiom is a proposition that defeats its opponents by the fact that they have to accept it and use it in the process of any attempt to deny it." The Law of Identity is an axiom; so is the Law of Contradiction. The concept of proof presupposes the existence of axioms from which such proof is derived. The laws of logic are the means by which one proves the truth of one's statements; the demand that one "prove" the laws of logic is a contradiction in terms.  ITOE

A philosophic axiom cannot be proved, because it is one of the bases of proof. But for the same reason it cannot be escaped, either. By its nature, it is impregnable. OPAR

 
It Is this skill which enables one to integrate the whole of their knowledge. The ability to recognize a misuse of language in the form of a contradiction.
 
The premise that something besides perception is required to validate axioms is true it is the ability to conceptualize the implicit and recognize a denial of the implicit as a meaningless contradiction.
 
The statement "How do I know 2+2 will always be 4?" is a claim that one could be aware of a entity that is not and entity.
 
You asked:
 

What is the referent of the number 1?

 
We form the concept one by abstracting from the concept entity.
 
Ms Rand answers this in ITOE in the appendix section called "Numbers"
 

[...]the term "one" is the concept "entity." And the concept "entity" is the base of your entire development. It has that great epistemological significance.
Prof. E: Is there a distinction in meaning or referent between the concept "unit" and the concept "one," in the sense, for instance, that you grasp that this is one ashtray, or one book? To regard it as a unit is to regard it, as you say, as a member of a class of similar things. Is that same perspective involved in grasping that it is one?
AR: Before you have a concept of numbers?
Prof. E: Yes. <ioe2_199>
AR: You will perceive that it is one, as an animal would, but you couldn't grasp the concept "one" without a concept of more than one—without a concept of numbers.
Prof. E: Perception gives you directly a certain kind of quantitative information.
AR: Yes.
Prof. E: Even prior to either implicit or explicit concepts.
AR: That's right.
Prof. E: And is it true that that quantitative information is presupposed, before you form even the implicit concept "unit"? In other words, a young child would have to perceive that this is one, even though it has no implicit concept of that, before it could even form the implicit concept "unit."
AR: Of course, and here is where we have to be Aristotelian: everything that exists is one. "Entity" means "one." But we couldn't have the distinction between what we mean by "one" vs. what we mean by "entity" if we didn't have the concept of numbers more than one which, after all, are only multiplied ones or divided ones.
Prof. E: So you get quantitative information by perception; then, via the process of grasping similarities and differences, you form the implicit concept "unit." You then rise to the general conceptual level, at which point you are able to form conceptually for the first time the concept of various numbers, including "one."
AR: Right.
Prof. E: Am I correct in saying that "one" and "many," as concepts, are metaphysical, while "unit," as a concept, is epistemological?
AR: That's right.

 

I have to stop here for now but I intend to get to induction soon.

 

Edited by Plasmatic
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Phew!  A lot to take in!  Thanks everyone for staying in this discussion!  It's very helpful to me and I'm glad to have a chance to discuss this stuff!

 

GrandMinnow, I'm not sure I explicitly disagree with anything you're saying.  I sense that you're just trying to stop wild speculation about the meaning of Godel's proof, which is admirable.  Your response to my initial statement did not appear to address it directly.  Do you believe that my initial statement (that Godel's proof defeated mathematical formalism) is true?

 

I agree with your later statement that it was not (as I had said ) "PRECISELY" what Godel proved, but requires a philosophical argument from the mathematical result.  But I believe that the dust has settled from several decades of debate, and there is something in the area of consensus about the above statement. 

 

About the clock where 2+2=0--yes, fine, but that's just a trick of words.  2 (with its usual meaning) plus 2 (with its usually meaning) equals four (with its usual meaning) though obviously if you change the rules governing the symbols, you can get any result you want. 

 

Plasmatic, I don't believe I have changed the way I'm defining a priori at all.  Interestingly, the 1st time I read Kant I actually thought he was doing the same thing you're accusing me of, and I wrote as much in the margin (because at one time he defined a priori as independent of experience, but than later said that experience causes it to arise).  I just didn't understand what he meant.  When I did, I saw that what he meant was consistent with both statements.  And all my statements that you quoted are consistent with each other, and what I mean.  A priori knowledge is that which...

  • Can't be proved or justified by sense experience
  • Allows us to conclude things beyond what is justified by sense experience
  • ...though sense experience can and does cause such knowledge to arise

 

I can see how you can think the 3rd statement contradicts the 1st two--partly because I once thought the same thing about Kant!  Maybe an example will help you see why I don't believe they're contradictory at all.  When I learn mathematics, I do so by means of a teacher drawing shapes on a board.  This allows me to see (in my "mind's eye" or with intuition) not only that the particular triangles she is drawing have certain properties, but that ALL TRIANGLES MUST HAVE THOSE PROPERTIES.  That is something that I didn't conclude by reasoning that,  "all the triangles my teacher drew had those properties, therefore ALL triangles have those properties."  That would be a fallacy (like concluding "all swans are white" from the fact that all the ones you've seen are so).  Rather, I see the necessity in my mind.  Thus, the sense experience (of watching my teacher draw it on the board) caused my knowledge to arise, but that experience does not justify my drawing a universal conclusion (like "all triangles have such and such properties")--the justification consists in something I saw with my minds' eye (or by intuition).  And all attempts to draw universal conclusions from limited sense experience is subject to the problem of induction.  Thus the universal conclusions we draw (in math for instance) must be based on something else. 

 

About Ms. Rand's quote on intuition--I think there are two sense in which the word "intuition" is used: (1) a vague feeling that something is true without being able to say why (i.e. "I had an intuition when I woke up this morning that such-and-such would happen"), and the other (2) is more technical, in the sense of the necessary connections we see between triangles in the examples above.  I don't believe Ms. Rand's definition of reason can be squared with Godel's proof.  Trying to explain how mathematical knowledge did not violate empiricism was exactly the goal of the formalist program that it defeated.  Specifically basing it on logic was the idea (and the topic of Russel and Whitehead's Principia Mathematica, and in the end that program failed because of the paradoxes, and was forever closed with Godel's proof).  That, anyway, is my understanding of it. 

 

I agree with what Ms. Rand says about intuition in the interview.  People often use "intuition" to describe very fast unconscious reasoning that we are not aware of--but in fact, it is often just plain old reason that's just not aware of itself.  But that's different than the a priori intuition I'm talking about.

 

Yes, what I mean by categories is very similar to what Ms. Rand means by "irreducible primaries."  But the way she defines existence (by sweeping one's arm around and saying, "I mean this") is not sufficient to define it for someone who does not implicitly recognize it.  Sweeping ones arms around is not sufficient to impart understanding of the meaning of the term "existence," and all of the "irreducible primaries" on which language rests are such that they cannot be defined by pointing as many "concretes" can.  Think the Helen Keller moment when she suddenly understands the idea that there is a connection between experiences of different kinds, and that such concepts ("kinds" if you will, and "kind" is itself an irreducible primary par excellence) have names.  You can't get that from the sense experience itself (since the only information given by sense experience is itself).  You can go on pointing to different things and naming them, but unless someone understands the concept of a "kind," they'll just be like poor Helen was prior to her awakening moment. 

 

I think the best Oist reply to this would be to say that you CAN get them by abstraction.  You abstract from sense experience to get to such irreducible categories (the Oist might say).  But abstraction itself requires a category.  To consider only the shape, or only the kind, or only the feel, or only the number, only this-or-that aspect of something, you have to have the concept of the aspect to-be-considered. 

 

Wow, I was actually shocked when I read this (quoting you quoting Ms. Rand): "Axiomatic concepts identify explicitly what is merely implicit in the consciousness of an infant or of an animal. (Implicit knowledge is passively held material which, to be grasped, requires a special focus and process of conscious-ness—a process which an infant learns to perform eventually, but which an animal's consciousness is unable to perform.)"

 

I'm not (from that, anyway) seeing any clear difference between what I am calling categories and what Ms. Rand is calling axiomatic concepts, or between what I am calling "a priori" and she is calling "implicit knowledge."  (Except perhaps that I wouldn't claim to know anything about what goes on in the minds of animals….)

 

If we have implicit knowledge, then all knowledge does not come from sense experience (some of it is implicit).  That's what I've been saying!

 

Ms. Rand's definition of an axiom that you gave works very well with the law of identity and non-contradiction in logic, but not so well in (for instance) mathematics, where there are also axioms which are not "propositions that defeat their opponents by the fact that they have to use it in the process of attempting to deny it." 

 

Furthermore, the empiricist program to show that mathematics was based on logic (or set theory) (a) failed to accomplish that, and (B) it was shown by Godel's proof that it is impossible to accomplish that. 

 

The interview at the end of your post gives a very brief window into Ms. Rand's view on how arithmetic is based on sensation,  but I don't see enough details to comment.  Personally I am skeptical (I may even go so far as to say I think its impossible).  I believe that it is more-like the "implicit knowledge" that she described. 

 

Your comment following that section about "getting to induction soon" was timely.  When I read Rand I felt that she didn't understand the problem of induction or its significance.  I also read a book by an Objectivist alleging to contain a solution to the problem of induction, and my sense was that the "solution" consisted of failing to understand the problem.

 

I saw your comment about Professort McCaskey "making inroads on the problem of induction."  I've tried to search out his idea, and I found that he makes the distinction of Socratic (as opposed to Scholastic) induction, and defines it  as "a compare-and-contrast process for discovering properties that characterize all members of a kind."  He makes the point that it goes from things (rather than individual statements) to universal terms (rather than universal statements).  From what I was able to follow, his idea is something like this: we start out not knowing what a thing is or its cause, but by constant experience we are able to narrow our definition of the concept.  Thus early experiences with cholera involved knowledge of certain conditions under which it was known to arise.  Eventually, human experiment was able to isolate the cause as certain type of bacteria, etc., in seeing more-and-more cases where the presence of the bacteria seems to lead to cholera.  Eventually it is isolated in a laboratory and scientists are able to directly observe the connection between the bacteria, and cholera. 

 

This seems like a straightforward and clearly reasonable description of something like the scientific method.  But the genius of Professor McCaskey (it seems to me) was in this.  To the Humean tradition which voices the objection, "yes, perhaps in the case observed by scientists on such-and-such an occasion, in the laboratory, and in the cases documents, in THOSE cases the bacteria led to the disease, but how do you know that it will always be so? --that tomorrow you won't wake up and find that eating apples causes cholera (etc.)…?"

 

His reply is, "THAT (the disease caused by the bacteria in question) is what I mean by cholera."

 

I think this is eminently reasonable, an excellent point, and one of the very few far-reaching attempts to solve the problem of induction that I've ever seen.  Moreover, I think something exactly like what he's describing  is actually true of how scientific conclusions can may come to constitute real knowledge despite the problem of induction.  Something like this is certainly a significant part of any solution to the problem of induction.  However, I see at least two reasons why (at least as far as my limited understanding goes) it doesn't solve the problem of induction:

 

  1. There are such a things as "natural kinds," and the scientific method in general, and the particular method proposed above presupposes as much.  There really is something called "cholera" and something called "vibrio cholerae" and those two things (a) are different, and (B) involve certain characteristics which everything that is that thing have in common (all true cases of cholera have certain things in common, etc.).  But the Humean question (how do you know that tomorrow apples won't cause cholera) has real significance here, and Professor McCaskey's reply constitutes something like a retreat.  It may actually be the case that the real thing called cholera is not actually caused by the bacteria in question, and simply saying "that's how I define cholera" may actually end up obscuring this fact. 
    1. Note: the situation is different in the case of simple mechanics, contrary to Hume, and in line with both Professor McCaskey and the Oist whose book I had read on the problem of induction, I believe we actually DO SEE the cause and its necessity (by what I've been calling "a priori intuition," even if that's not what the Prof and Ms. Rand prefer to call it--I will accept it under any name!). 
  1. There are many scientific conclusions where the concepts themselves are not the things in question.  For instance, some such conclusions describe relationships where the concepts are well-understood and agreed to, but the mathematical relationships among them are the topic of the conclusions.  So for example, an equation about the force of gravity purports to describe the actual movement of planets in measurable terms.  The terms are not in question; the measurements are.  So when the Humean asks, "how do you know tomorrow that gravity will continue to exert a force inverse to the square of distance?"--it doesn't really work to just say, "well that's how I define gravity" (or worse, since gravity predicts certain positions, to say, "that's how I define distance," etc.).  That would be like (ahem) claiming that 2+2=0 on the basis of a clock made that way.  It is still possible to ask, "how do you know tomorrow that gravity will have such a force?"--and unless (like in simple mechanics) we actually see the necessity in it (as we do in simple mechanics), we have to admit that we're assuming something that is not justified.  Incidentally, I believe we DO see the necessity in the inverse square law that defines gravity (or something very close to it). 

 

But in my view, the problem of induction is much-worse than this, and any enquiry serious enough to be called science depends on a number of ideas that cannot be justified in this way, or in any other way but by a type of knowledge that cannot come from sense experience.    

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I should point out that GrandMinnow said,

 

"As I mentioned, that is not what Godel proved. And it is not well put to say that the relative consistency and independence of the continuum hypothesis is an implication of the incompleteness theorem. And I don't see how the relative consistency and independence proofs refute a refutation of formalism. By the way, speaking of the independence of the continuum hypothesis, proved by Cohen, we note that Cohen considered himself a formalist."

 

I never made such statements nor did I intend to imply them. However, GrandMinnow was correct in asserting that I switched the results of Cohen and Godel. Mea Culpa.

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Perhaps what you have in mind is that Godel's second incompleteness theorem is that no recursive, consistent, arithmetically-adequate theory can prove its own consistency. This has nothing to do with "a system proving itself" nor does it even entail that there are not axiomatic systems that don't prove their own consistency.

Ah, yeah it is. "Prove itself" is meant to be proving its own consistency in this sense. I am not trying to say no axiomatic system can prove its -consistency- alone.

 

 

Godel did have philosophical follow-ups to his mathematical results, but I don't know what writings by Godel or in the philosophical literature you have in mind.

I was paraphrasing JM's argument. It's not mine. I agree with you. I also typoed; it should say "not only".

Edited by Eiuol
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John, we are making progress:

 

John said:

 

 

This seems like a straightforward and clearly reasonable description of something like the scientific method.  But the genius of Professor McCaskey (it seems to me) was in this.  To the Humean tradition which voices the objection, "yes, perhaps in the case observed by scientists on such-and-such an occasion, in the laboratory, and in the cases documents, in THOSE cases the bacteria led to the disease, but how do you know that it will always be so? --that tomorrow you won't wake up and find that eating apples causes cholera (etc.)…?"

 

His reply is, "THAT (the disease caused by the bacteria in question) is what I mean by cholera."

 

I think this is eminently reasonable, an excellent point, and one of the very few far-reaching attempts to solve the problem of induction that I've ever seen.  Moreover, I think something exactly like what he's describing  is actually true of how scientific conclusions can may come to constitute real knowledge despite the problem of induction.  Something like this is certainly a significant part of any solution to the problem of induction.

 

 

 

Recall what I said previously:

The fact is, the question "how are we justified in drawing the universal conclusion that 1+1 will ALWAYS equal two?" is answered by knowing what in experience the symbols you are referring to mean.

 

That is, the statement "1+1 may not always equal 2" is a nonsensical missuse of language. That is, it is a contradiction. Logic is the method of preserving meaning via linguistic-conceptual means. The meaning of a concept is its referents. The contradiction of philosophic primaries is known- understood by reducing the symbolic-linguistic-conceptual tools to their origin in experience and thereby identifying the conditions that satisfy and establish the correspondence of the statement.-----

 

It Is this skill which enables one to integrate the whole of their knowledge. The ability to recognize a misuse of language in the form of a contradiction.
 
The premise that something besides perception is required to validate axioms is true it is the ability to conceptualize the implicit and recognize a denial of the implicit as a meaningless contradiction.
 
The statement "How do I know 2+2 will always be 4?" is a claim that one could be aware of a entity that is not and entity.

 

 

 

And in the quote I posted previously Ms. Rand said:

 

 

to define "existence," one would have to sweep one's arm around and say: "I mean this."

 

The whole discussion is about concepts and what the cognitive use of them is. We use words-language to convey and preserve meaning. The symbols are a means of achieving referential economy and integration. Context, hierarchy and integration are all methods of connecting extrospective and introspective experiences into a cognitive whole. The whole need of the general science of epistemology is to teach men how to do this explicitly precisely because man is capable of contradiction.

 

 

man's life span is a continuum whose only integrator is his conceptual faculty.

The Ayn Rand Letter Vol. II, No. 16  May 7, 1973 The Missing Link

 

 

Justification, proof, reference and definition are all matters of meaning. That is, they all are devices for conveying, recalling and integrating the content of perceptual experience (introspection included) for the purpose of survival. We reduce a concept back to its origins in experience-reality because it is perceptual experience that we are trying to recall and convey so as to situate ourselves in relation to reality in a life preserving manner. The connection to existence for man starts with perception. All philosophic primaries are implicit in perception at any moment of awareness.

 

Being implicit from the beginning, existence, consciousness, and identity are outside the province of proof. Proof is the derivation of a conclusion from antecedent knowledge, and nothing is antecedent to axioms. Axioms are the starting points of cognition, on which all proofs depend. OPAR

 

There is no perception of something that does not exist, have identity or by other means than consciousness. One only ever perceives entities. There are no unextended entities in perception. There are no floating actions apart from entities in perception. Every state of awareness contains a multiplicity of bounded particulars-entities.

 

The "problem" of induction has been heavily interlaced with the "problem" of universals and your comments about natural kinds seem to evidence some knowledge of this. Both "problems" are about concepts and the attempt to grasp what they are and how we use them.

 

John said:

 

 

Note: the situation is different in the case of simple mechanics, contrary to Hume, and in line with both Professor McCaskey and the Oist whose book I had read on the problem of induction, I believe we actually DO SEE the cause and its necessity (by what I've been calling "a priori intuition," even if that's not what the Prof and Ms. Rand prefer to call it--I will accept it under any name!). 

 

Is the "other Oist" you are referring to David Harriman? Its worth mentioning that there are differences between Prof. McCaskey and the view of induction in Harriman's book that contains Dr. Peikoff's theory of induction.

 

Here are some topics relevant to this:

http://forum.objectivismonline.com/index.php?showtopic=27504&hl=induction

 

http://forum.objectivismonline.com/index.php?showtopic=20690&hl=induction#entry258241

 

http://forum.objectivismonline.com/index.php?showtopic=20086

 

An interesting fact is that in the last link Grames and I were debating the very differences implicit in Prof. McCaskey's and Dr. Peikoff's views only days before the disagreement was made public.

 

The contentions seem to be centered on an ambiguity about two functions of concepts. The first is the role I have been discussing in this thread in relation to referential economy.

 

The second is related to your comment:

 

 

[...]the Humean question (how do you know that tomorrow apples won't cause cholera) has real significance here, and Professor McCaskey's reply constitutes something like a retreat.  It may actually be the case that the real thing called cholera is not actually caused by the bacteria in question, and simply saying "that's how I define cholera" may actually end up obscuring this fact [...]

 

For now I will just post this quote and return in a bit:

 

Axiomatic concepts are the constants of man's consciousness, the cognitive integrators that identify and thus protect its continuity. They identify explicitly the omission of psychological <ioe2_57> time measurements, which is implicit in all other concepts.

 

It must be remembered that conceptual awareness is the only type of awareness capable of integrating past, present and future. Sensations are merely an awareness of the present and cannot be retained beyond the immediate moment; percepts are retained and, through automatic memory, provide a certain rudimentary link to the past, but cannot project the future. It is only conceptual awareness that can grasp and hold the total of its experience—extrospectively, the continuity of existence; introspectively, the continuity of consciousness—and thus enable its possessor to project his course long-range.

[...]

 

AR: Consider all the people born in the eighteenth century, let us say—men who couldn't possibly be alive today. When you use the word "man" in reference to them, the concept "man" stands for existing men, even though they do not exist now. The meaning of a concept includes, as I have said repeatedly, not only all the present referents but also all the future ones that anyone might consider, and all the past instances. The meaning remains the same, the nature of the referents remains the same, that which they have in common with present men (and which made you include them in the concept "man") remains the same, even though the particular physical concretes are not there any longer.

 

Prof. D: Well, I certainly agree to that. But if you equate the meaning of the word with the existents, if you make that theoretic move—

 

AR: The meaning of the word includes all the instances of that existent. Specifically and emphatically not only the presently existing referents, but all the referents of that kind, past, present, and future. If a concept did not do that, <ioe2_177> it would not be a concept—you could form it today, but you could not use it tomorrow, and you could not use it to think about yesterday. [...]

 

Mathematics is the science of measurement. Before proceeding to the subject of concept-formation, let us first consider the subject of measurement.[...]Now what is the purpose of measurement? Observe that measurement consists of relating an easily perceivable unit to larger or smaller quantities, then to infinitely larger or infinitely smaller quantities, which are not directly perceivable to man. (The word "infinitely" is used here as a mathematical, not a metaphysical, term.) The purpose of measurement is to expand the range of man's consciousness, of his knowledge, beyond the perceptual level: beyond the direct power of his senses and the immediate concretes of any given moment. Man can perceive the length of one foot directly; he cannot perceive ten miles. By establishing the relationship of feet to miles, he can grasp and know any distance on earth; by establishing the relationship of miles to light-years, he can know the distances of galaxies.

 

The process of measurement is a process of integrating an unlimited scale of knowledge to man's limited perceptual experience—a process of making the universe knowable by bringing it within the range of man's consciousness, by establishing its relationship to man.

 

ITOE

 

The historical debate in the philosophy of science over observation language and theoretical language is centered on this aspect of conceptualization. In particular the conceptualization of unobservable causes...

Edited by Plasmatic
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[...] GrandMinnow cleaned the floor with all of us [...]

I didn't mean to do that. My main point is that incompleteness and some other topics mentioned here are technical mathematical results, so one needs to properly study the actual mathematics before using this mathematics as a basis for philosophical claims. I don't deny that there is a rich philosophical discussion that comes in the wake of the mathematics, but for these discussions the mathematics must first be carefully understood. Then, putting the matter in Objectivist context further complicates this, since the mathematics itself is not devised to necessarily comport with the Objectivist framework.

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[...] GrandMinnow said,

 

"[...] it is not well put to say that the relative consistency and independence of the continuum hypothesis is an implication of the incompleteness theorem. And I don't see how the relative consistency and independence proofs refute a refutation of formalism. [...]"

 

I never made such statements nor did I intend to imply them. 

Then I don't follow what you meant by "implies the opposite" if not to say that Molineux's proposed refutation of formalism is rebutted, and "for example" if not to say that the independence of the continuum hypothesis is part of what is implied by incompleteness in the passage:
 
"You say, "Mathematical formalism [...] was defeated by Kurt Godel's incompleteness proof." It seems that Godel's Incompleteness Theorem implies quite the opposite of what you are saying. For example, Godel proved that you cannot prove the Continuum Hypothesis." [alpeh_1]
 
But if I have misunderstood you then I stand corrected, especially since my point is not to pin anyone down here but rather to keep straight the mathematics discussed and to suggest restraint in drawing philosophical conclusions based incorrect or vague understandings of the mathematics. In any case, it seems wrong to say that the incompleteness theorem does the "opposite" of refuting formalism. If it is correct to claim that the incompleteness theorem weighs in at all about formalism then it does so against formalism and not in favor of it.
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Minnow said:

I didn't mean to do that. My main point is that incompleteness and some other topics mentioned here are technical mathematical results, so one needs to properly study the actual mathematics before using this mathematics as a basis for philosophical claims. I don't deny that there is a rich philosophical discussion that comes in the wake of the mathematics, but for these discussions the mathematics must first be carefully understood. Then, putting the matter in Objectivist context further complicates this, since the mathematics itself is not devised to necessarily comport with the Objectivist framework.

For the same reason that I do not consider your comments to me to be a wiping of anything, I'll point out that the entire discussion of technical mathematics and Godel is misguided by Oist standards. One does not use any special science to reach backwards to the foundations upon which it can have no say whatever according to Objectivism.

Math is a language and follows all the same epistemic prescriptions of a valid epistemology. My statement about Godel was actually a question about John's associating it with philosophical foundations-axioms.

Edited by Plasmatic
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Do you believe that my initial statement (that Godel's proof defeated mathematical formalism) is true?

I don't have a firm view on the matter. But I think it is safe to say that if incompleteness weighs in at all about formalism then it does so in disfavor of formalism and it's hard to imagine that it does so in favor of formalism. But again I caution that this depends on a more precise understanding of formalism and should not depend on common misconceptions about what formalism is. In any case, in the specific sense I mentioned - Hilbert's proposal for a finitistic proof of the consistency of the ideal mathematics of set theory, of analysis, and even of arithmetic - the incompleteness theorem would seem to defeat the proposal, and as I understand, this accords with a consensus of informed writers on the subject, though there is still some debate on the technicalities and even some informed dissenters. 

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 Trying to explain how mathematical knowledge did not violate empiricism was exactly the goal of the formalist program that [the incompleteness theorem] defeated.  Specifically basing it on logic was the idea (and the topic of Russel and Whitehead's Principia Mathematica, and in the end that program failed because of the paradoxes, and was forever closed with Godel's proof). 

 

[....]

 

Furthermore, the empiricist program to show that mathematics was based on logic (or set theory) (a) failed to accomplish that, and ( B) it was shown by Godel's proof that it is impossible to accomplish that. 

 

I don't know what specific writers you have in mind when you mention "the empiricist program". In whatever sense Hilbert's views can be described in terms of empiricism, perhaps the most salient aspect of his view was in the distinction between the contentual (which is finitistic) and the ideal (such as abstract set theory with infinite sets) and the proposal to use only finitistic mathematics to prove the consistency of ideal mathematics. 

 

Russell though is famous for his own version of empiricism in general and logicism in mathematics. Principia Mathematica (PM) has not been vitiated by any paradoxes, as far as I know. Rather, it was Frege's system that was shown to be inconsistent by Russell, and PM was devised by Whitehead and Russell as a remedy. Then Godel showed PM to be incomplete, but he did not show it to be inconsistent (anyway, trivially, any inconsistent theory is complete). Logicism is another matter. Whatever we may say of logicism, it is commonly asserted that PM itself does not do the job since PM uses three axioms that one would be inclined to say are not logical: infinity, choice, and reducibility, as even admitted by Russell. However, even on that point there is debate on technicalities and some informed holdouts who propose their own versions of logicism. 

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I think the best Oist reply to this would be to say that you CAN get them by abstraction.  You abstract from sense experience to get to such irreducible categories (the Oist might say).  But abstraction itself requires a category.  To consider only the shape, or only the kind, or only the feel, or only the number, only this-or-that aspect of something, you have to have the concept of the aspect to-be-considered. 

Can you explain to me 1) what you mean by concept, and 2) why a category doesn't also require a category.

 

Regarding my 1, concept I've seen refer to any form of an idea, or even some type of idea prior to your awareness as in the case of what I described as "a priori" mechanisms. I see Plasmatic has many quotes, but for what it's worth I don't usually agree with his understanding when it comes to topics pertaining to epistemology.

 

Regarding my 2, how you think of a category will be affected by how you think of concepts. Objectivism treats concepts as knowledge. "Implicit knowledge" is a good term, but it isn't at all really what someone knows in the category sense. It is what is implied by a child's knowledge for instance, but only an adult knows. A child may well know what a tree is, but lack a concept of existence. Knowing what a tree is means having the concept "tree", and that implies existence, perhaps in a logical hierarchy. Developmentally, "tree" is first, not "existence". An a priori category is something at root of a hierarchy in what a person must have "in their head" somehow or else they won't be able to know anything. But I reject that type of basis/foundation as valid.

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Plasmatic, your claim that we are justified to draw the universal conclusion that 1+1 will ALWAYS = 2 is "answered by knowing what in experience the symbols you are referring to mean," is, in my view, a significant source of our remaining disagreement.  I actually thought the same thing at one time (again, the 1st time I read Kant I write something almost exactly like that in the margin).  But for the following reasons, I now believe this cannot be the case. 

 

  1. It has been shown to be impossible to derive all of mathematics from logic.  The 20th-century attempt to do so by deriving all theorems from set theory ended up proving that there are theorems that cannot be proven (and moreover, it is subject to paradox such as Russell's paradox). 
  2. Most famously, Godel's theorem proved that in any consistent mathematical system rich enough to contain arithmetic, there exist propositions that cannot be proven (or disproven) from the system's axioms--that is to say, there are propositions that are NOT simply a matter of (as you say) "knowing what the terms mean."

 

I believe GrandMinnow (who is apparently seen as the resident Godel-interpretation expert) will attest to both these points, probably adding that "a philosophical argument is needed to draw the latter part of my conclusion on point #2."  =)

 

But I believe the burden of proof is rather light here, and the needed-philosophical arguments have been supplied and reached something like consensus among most experts.  Rather, a philosophical argument is needed to say that our knowledge of mathematics is based on "knowing what the symbols mean," and such an argument has to contend with the whole 20th century, in which many of the world's most brilliant minds tried, and failed, to provide JUST such an argument, and were eventually silenced by Godel's proof. 

 

Yes, the Oist I am referring to is David Harriman. 

 

Thanks for the links on Prof McCaskey's discussions on the problem of induction.  I plan to follow up with that stuff for sure!

 

Minnow, when I mention the defeat of the "empiricist program," I am talking about such writers as Wittgenstein, Carnap, and the logical positivist camp, for whom the attempts to ground mathematics in logic / set theory was part of an attempt to settle one important part of the larger empiricist-rationalist debate.  In order to maintain that all knowledge comes from sense experience, you have to explain mathematics in a way that is consistent with that.  Since obviously sense experience itself can't give you the universal statements of mathematics directly, the ideas that it was simply a matter of the extrapolation of axioms is the only suggerstion I've ever heard argued.  (Granted, I am not an expert in mathematical logic, and I'm confident that, as you say, "there are some informed holdouts.") 

 

For my part, there seems to me to be a direct path as follows:

 

  1. in any consistent mathematical system rich enough to contain arithmetic, there exist propositions that cannot be proven (or disproven) from the system's axioms.

...to…

  1. Some mathematical propositions of which we have knowledge do not consist of the mere extrapolation of concepts

...and since…

  1. Mathematical knowledge cannot come directly from sense experience (which only consists of itself)

...and…

  1. We have mathematical knowledge

...we can conclude…

  1. Some of our knowledge does not come from sense experience

 

I believe I could put that all in syllogisms that would be very difficult to assail (I'll do so if requested). 

 

Merry Christmas everyone!

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Eiuol,

 

What I mean by a concept: I almost want to say that by concept I mean what everyone means by concept!  It is itself a category, so it can't be defined without synonyms (which include "kind," type," maybe even "property," etc.).  I think if I were forced to define it I may say something like "something that different things have in common." 

 

Why a category doesn't also require a category: The thing I think we know is that SOME concepts have to be foundational.  I don't believe that they are reasoned to with a cosmological argument-type way where the first premise is, "all concepts can be understood by other concepts."  If that were the case, a category would (as you say) also need a category.  We simply observe that most words are understood by other concepts.  We then ask ourselves, "are all words this way?"  And if all words were understood by other concepts, there would only be two possibilities:

  1. Infinite concepts (clearly not the case)
  1. A circular structure "unattached to understanding" (clearly not the case, and also if it were, understanding would be impossible)

 

I take your point about a child knowing a tree before the concept of existence.  But as Ms. Rand says, the knowledge of existence may be "implicit."  

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