Discuss:Mathematics, Logic, Philosophy:Axiomatic Method
From da Vinci Concept
[poll] Axiomatic Method
henrik - Sun Feb 8 7:24:27 2004
"The only thing that is objective is rigorous axiomatic mathematical proofs, and I have some doubts about that."
This quote is taken completely out of context from one of charmed_quark's postings. However, I could not resist the temptation to make some sort of comment about it.
The Axiomatic Method is often taken to be THE most rigorous form of knowledge acquisition. You start out with a small set of axioms. These axioms determine the ontology you are going to talk about and the methods by which new information can be acquired about this ontology.
The oldest exposition of the Axiomatic Method known to us is Euclid's Elements, which laid the foundations for all of our knowledge of Euclidean Geometry.
Euclid was way ahead of his time, quite literally by a couple of millennia.
It was only until the beginning of the twentieth century that mathematics was given a solid axiomatic foundation.
This was prompted partly by David Hilbert's program as one of the challenges for mahematicians for the 20th century. But also, the work done by George Cantor on infinite sets and some of the logical contradictions that it led to had shown how mathematics was in desperate need for a solid foundation.
It took a while before such a foundational system could be found and then some more time for it to be accepted by the mathematical community at large. However, it is now accepted that *all* of mathematics can be derived from ZFC, also known as Zermelo Fraenkel Set Theory.
Now back to the quote from charmed_quark...
Is it fair to call knowledge generated by the Axiomatic Method as "objective"?
Hmm...
By "objective" we usually mean something that only depends on the object itself and not on the subject that is interacting with it.
At one level, formal mathematics can be said to be objective in the sense that no matter which mathematician you talk to they will all have to come to agree with each other if they are using the same axioms. Thus, the information that is gathered about the Natural Numbers is independent of the subjects and only depends on the object itself, in this case the Natural Numbers.
And of course, mathemtical truths being objective is a good thing. Imagine if one mathematician could prove that there are infinitely many twin primes and another could prove that there are only finitely many of them.
Although, this leads us into a somewhat ackward positon...
If mathematical truths are objective, what kind of objects are they? We can't grab them with our hands and squish them. Where do these mathematical objects live? Do they live in some sort of Platonic realm? Are they just some mental construction? And if they are just a mental construction, how do I know that my concept of the real numbers is the same as your concept of the real numbers? After all, my concept lives inside my mind to which you do not have access to. And if mathematical concepts live in peoples minds doesn't this seem to suggest that these objects are in fact subjective?
Henrik.
--henrik Sun Feb 8 7:24:27 2004
sunny - Sun Feb 8 12:06:12 2004
I believe that the poll is faulty. It assumes a false dichotomy between analytic truth and synthetic truth.
However, firstly, it would be prudent to place a reference for "Platonic Realm" and perhaps provide other references that speak to the questions that you are posing. Although I may have been introduced to these ideas, a simple and well written exposition would create context that better enables discussion.
I am going to hold to Objectivist epistemology in my statements.
Mathematical objects, like ANY human knowledge, are conceptual. Conceptual objects exist within the mind as referents to that which is observable, either directly, or through a process of derivation and abstraction, indirectly. Mathematical objects are therefore neither exclusively within the mind nor exclusively external to the mind. Your poll does not even allow for these two choices, as a Platonic Realm is not a place that is 'out-there' as in the sense of reality, but 'out-there' in the sense of some unreal and untouchable super-reality that is independent of my *supposedly* faulty means of perception or cognition. Concepts do not exist in a Platonic Realm. They are existents of my conciousness.
Contrary to Platonism, concepts are formed upon the basis of directly observable 'primary' existents, real objects. Concepts are referents to these existents. Concepts live within the minds of humans, and the are referents to other existents. The basis of mathematical knowledge, like all other human knowledge, is that which is observable to our senses and the tools that we create to expand those senses, or is derivable from those observations. Meaning that mathematical knowledge, like all other true knowledge, is objective. The differences between my understanding of a mathematical concept and yours (you being a Mathematician) is only a matter of our conceptual development and degree of specificity.
I will refer to Introduction to Objectivist Epistemology page 88, "The Analytic-Synthetic Dichotomy" by Leonard Peikoff. This source deals in detail with the problem of objectivity in Mathematics, particularly the analytic-synthetic dichotomy.
A quote taken from the conclusion of this work identifies the epistemological results of the analytic-synthetic dichotomy:
| The ultimate result of the theory of the anlytic-synthetic dichotomy is the following verdict pronounced on human cognition: if the denial of a proposition is inconceivable, if there is no possibility that any fact of reality can contradict it, i.e., if the proposition represents kownledge which is certain, then it does not represent knowledge of reality. In other words: if a proposition cannot be wrong, it cannot be right. A proposition qualifies as factual only when it asserts facts which are still unknown, i.e., only when it represents a hypothesis; should a hypothesis be proved and become a certainty, it ceases to refere to facts and ceases to represent knowledge of reality. . . .This theory represents a total epistemological inversion: it penalizes cognitive success for being success. |
| -- The Analytic-Synthetic Dichotomy, Leonard Piekoff, 1967, page 118 of Introduction to Objectivist Epistemology |
To the Poll I would add another option that is critical:
Objective concepts that are referents to existents
Existent meaning: existents of conciousness (other concepts) or physical existents, ostensively identifiable or through scientific means.
I think that the following are a true statements.
Mathematical objects live in the minds of humans as referents to existents of conciousness.
Mathematical objects are valid concepts when they have valid underlying premises based on objective reality. Because mathematics is a very rigorous field of knowledge, it is rarely the case that a mathematical object is not human knowledge (is not based upon valid premises).
Some related references:
IOE, The Analytic-Synthetic Dichotomy, by David Ross
Analytic/Synthetic - (A reply to Michael Heumer) (Discusses Analytic-Synthetic Dichotomy with respect to intrinsicism and subjectivism.)
Introduction to Objectivist Epistemology - Objectivism Reference Center
[ Edited Thu Feb 19 2004, 10:33PM ]
--sunny Sun Feb 8 12:06:12 2004
henrik - Mon Feb 9 12:23:15 2004
| I believe that the poll is faulty. It assumes a false dichotomy between analytic truth and synthetic truth. |
| -- sunny |
Ahh... but it doesn't! I make no assumptions about whether the classification of truths into analytic and synthetic is justifiable or not.
The dicussion about analytic/synthetic distinction is a fascinating subject onto itself. And although it is related to the ontological status of mathematical concepts, I doesn't seem to be immediately relevant to the discussion at hand.
| Your poll does not even allow for these two choices, as a Platonic Realm is not a place that is 'out-there' as in the sense of reality, but 'out-there' in the sense of some unreal and untouchable super-reality |
| -- sunny |
By *Platonic Realm*, I was not excluding what you just described. Actually, I believe that a Platonic Realm can be best thought as an untouchable super-reality accessed through our thoughts.
| I am going to hold to Objectivist epistemology in my statements. Mathematical objects, like ANY human knowledge, are conceptual. Conceptual objects exist within the mind as referents to that which is observable, either directly, or through a process of derivation and abstraction, indirectly. Mathematical objects are therefore neither exclusively within the mind nor exclusively external to the mind. |
| -- sunny |
Now, this is more like it. This is directly relevant to the discussion.
Let me paraphrase what you said just to make sure that I understood what you said.
Mathematical objects are conceptual objects and are "observed" through a process of derivation and abstraction.
Mathematical objects live in the minds of human beings but they depend on the external world since the process of abstraction usually starts with observations of the external world.
However, this seems to say that Mathematics and what is considered to be a mathematical truth is simply a social construction. You could even push this further and generalize it to all conceptual objects mathematical or not.
Is truth just a social construction that happens to have the property of being a reliable predictor of diverse phenomena in the world?
[ Edited Sat Feb 14 2004, 12:31PM ]
--henrik Mon Feb 9 12:23:15 2004
sunny - Thu Feb 19 19:20:59 2004
So you would also say that all knowledge is a 'social' construction? This seems very flawed. Taxes are a 'social' construction. Public education is a 'social' construction. Mathematics is an epistemological construction and a very rigorous one at that.
Concepts relating to the interactions between human and human deal with social knowledge. Human knowledge is also not arbitrary or subjective, but unlike concepts of other existents it just so happens that human social knowledge did not have to be as it is but might have been otherwise. Human history could have been otherwise. Human knowledge that concerns human interactions in this way are constructed. It is knowledge fabricated by humans for the purposes of humans.
Mathematics is deducted. It is taken directly from reality. Our perceptions of reality are valid and real. Mathematics is also valid and real. It is not taken from God, the Supreme Fascist, or from the whim of the Mathematician.
| Is truth just a social construction that happens to have the property of being a reliable predictor of diverse phenomena in the world? |
| -- henrik |
Absolutely not. You are confusing 'constructed' knowledge (that which is fabricated by man that could have been otherwise; social conventions, social constructions, fiction, fantasy) and perceived knowledge (that which is known from reality and that which can be conceptually abstracted from what is perceived).
--sunny Thu Feb 19 19:20:59 2004
sunny - Thu Feb 19 19:29:12 2004
Please reply to the above. But I have another topic that may interest you.
I am going to write up a paper on a possible 'solution' to the Incompleteness Theorem. That is, a solution in the sense that I may have come up with a means to repair, quite consistently, the logical ambiguities in Mathematics and other formal logical systems. The idea is fairly simple, but I must do a little more research into the area to discover what work that has already been done before disclosing it. I'll share it with you directly upon my visit to Amsterdam this March.
:)
--sunny Thu Feb 19 19:29:12 2004
henrik - Fri Feb 20 18:14:47 2004
| Mathematics is deducted. It is taken directly from reality. |
| -- sunny |
I am not sure I understand what you are saying. How is it that Mathematics is taken directly from reality? What do you mean by that?
| Is truth just a social construction that happens to have the property of being a reliable predictor of diverse phenomena in the world? |
| -- henrik |
| Absolutely not. You are confusing 'constructed' knowledge (that which is fabricated by man that could have been otherwise; social conventions, social constructions, fiction, fantasy) and perceived knowledge (that which is known from reality and that which can be conceptually abstracted from what is perceived). |
| -- sunny |
That is exactly my point. My current view is that Mathematics is a social construction in the same way that fictional stories, programming languages, video games and game boards like monopoly are social constructions. It is a very powerful social construction with many applications and a long history.
The process of conceptual abstraction from what is being perceived, is de facto the process of finding what kind of constructions seem to capture the relevant properties of the phenomena being perceived.
--henrik Fri Feb 20 18:14:47 2004
henrik - Fri Feb 20 18:43:43 2004
I am not sure what logical ambiguities you are referring to.
A view that is hold by many today is that although Mathematics is Incomplete, this is not really important for Mathematics.
Solomon Feferman seems to take the view that if a 'mathematical statement' is undecidable then it really is not a "mathematical question".
At this time, I do not know whether to agree or to disagree with this view.
Although, my tendency woould be to disagree.
I'll be curious to hear what you have to say.
Best,
Henrik.
--henrik Fri Feb 20 18:43:43 2004
sunny - Fri Feb 20 19:05:17 2004
| Mathematics is deducted. It is taken directly from reality. |
| -- sunny |
By this I mean exactly what I am saying. Mathematics, like all concepts, at its base, is observed directly: the concept of counting and numbers and their abstraction to larger and larger numbers. The concept of sets. These mathematical concepts are abstracted directly from observation.
I know that I can observer directly the basis of mathematics because I can integrate several similar objects that I perceive: these are all chairs, these are all pencils (this is counting.) I can also differentiate between objects that differ in their characteristics: chair is not a table but they are furniture, pencil is not a pen but they are writing instruments (these are sets.)
Concepts, valid concepts, are always, in their basis, mathematic. The fundamental difference between the mathematical concept of number and the conceptual differentiation and integration process is one of abstraction. The concept of number (1, 2, 3, pi) is abstracted from the process through which we integrate and differentiate real objects.
There is simply nothing mystifying about it. Mathematics is not fantasy. I can count the number of words in this reply. That is as closely tied to reality as you ever need to be. I would state that because mathematics is so simple in its basis, that it is more closely tied to reality than other complex concepts. It is for this very reason that pure mathematics works to find new answers. Mathematics is not just self-consistent. It is reality-consistent.
I implore you to read on the Objectivist theory of concepts [ Introduction to Objectivist Epistemology, by Ayn Rand ]
--sunny Fri Feb 20 19:05:17 2004
sunny - Fri Feb 20 19:17:41 2004
In reply to your poll
Mathematical objects exist within reality. Mathematical concepts, that refer to mathematical objects exist within the human mind.
Mathematical objects that exist within the human mind as abstractions exist within THIS reality. They are directly observable by your concious mind.
Not 'Platonic Realm'
Not 'Within the minds of humans'
Not 'Neither'
Both concious existence AND concrete existence
Both within the human mind AND within perceived reality
If I were to choose any reply from the poll I would either NEGATE the human minds involvement:
'Both within the human mind AND within perceived reality'
(choose 'neither') TRUE AND FALSE == FALSE
or I would IGNORE the necessary component of external reality, that which the human mind involves itself with:
'Both within the human mind AND within perceived reality'
(choose 'within the minds of humans') TRUE AND FALSE == FALSE
The 'within perceived reality' is not a 'DO NO CARE' - it is necessary to ensure that Mathematics, unlike fantasy, is not an arbitrary construction.
[ Edited Sun Feb 22 2004, 03:03PM ]
--sunny Fri Feb 20 19:17:41 2004
charmed_quark - Sat Feb 21 14:52:50 2004
Mathematical objects exist within reality?
If you have a 'one apple, two apple type' problem it is easy to think that the math comes from observed reality. But as soon as we take a slightly more complicated problem, say, determining the length of the sides of a right triangle, things become more disconnected from reality. The only thing that I'm counting is some units of length that I made up. And no matter what units I make up, I will have a hard time being able to count all three sides of the triangle with good ol' natural numbers (except for special case of 3,4,5).
What I'm trying to point out is that mapping math to reality requires quantification. That many real concepts elude quantification suggests that maybe math is not a fundamental principle of the universe.
Does math make for a useful model of some things? Sure. Does this mean that math actually reflects objective reality? Not necessarily.
--charmed_quark Sat Feb 21 14:52:50 2004
sunny - Sun Feb 22 11:54:58 2004
| What I'm trying to point out is that mapping math to reality requires quantification. |
| -- charmed_quark |
Are you implying that a process of quantification in measurement is not a process of observing objective reality?
Is it necessary that mathematics refer solely to objective reality? No.
It can also apply to human constructed systems for which the underlying premises of the application of formal logic apply. Different rules for constructed systems.
Take a step back from the question in hand. It is mathematics itself that we are speaking of. Mathematics itself is something derived directly from fundamental perceptual observations of objective reality. These are the concepts of 'existence' and 'identity'. Mathematics becomes more complex by using the concept formation process of differentiation and integration of objects that are observed. That is; observed from perception of concrete reality OR observed from introspection into existents of conciousness.
I implore you to acquaint yourself with Objectivist Epistemology and the objectivist theory of concepts.
[ Edited Sun Feb 22 2004, 02:58PM ]
--sunny Sun Feb 22 11:54:58 2004
sunny - Sun Feb 22 12:39:14 2004
Regarding the actual topic of discussion, the Axiomatic Method.
I think that we need to make a distinction between mathematics, the science, that is objective, and the various applications of mathematics as it is used within other sciences and human constructions.
I think that the Axiomatic method is both a method for creating a mathematic science and in creating instances of formal theory in application of mathematics. Depending upon the axioms chosen you may use the Axiomatic Method to determine the outcome of any statements taken to be axiomatic or primary. I think that this is what enables the tools of mathematics, the science, to be so versatile.
It is when the Axiomatic Method is applied to mathematics, the science and when these axioms are themselves self evident from objective reality; that mathematics, the science, is objective truth. In this case the axioms must be observable. They are not whims of the mathematician.
--sunny Sun Feb 22 12:39:14 2004
henrik - Sun Feb 22 17:30:28 2004
| Does math make for a useful model of some things? Sure. Does this mean that math actually reflects objective reality? Not necessarily. |
| -- charmed_quark |
I agree.
And this is exactly what seems so intriguing.
On one hand, it is very tempting to think of the theorems of mathematics as being a reservoir of universal objective truths that somehow directly reflects reality.
On the other hand, mathematics seems to be a mental game. We choose some axioms that seem plausible and we play around with these axioms using logic to investigate what kind of consequences we can derive from these axioms.
| Regarding the actual topic of discussion, the Axiomatic Method.[...] It is when the Axiomatic Method is applied to mathematics, the science and when these axioms are themselves self evident from objective reality; that mathematics, the science, is objective truth. In this case the axioms must be observable. They are not whims of the mathematician. |
| -- sunny |
You say that mathematics represents objective truth as long as we choose axioms that are self-evident from objective reality. This seems to be reasonable...
However, how do you choose these axioms?
Let me give a concrete example.
The Axiom of Choice is the following:
Given any collection of non-empty sets, there exists a set A, which contains exactly one element from each one of these non-empty sets.
For example, suppose you have a collection of pairs of socks. The axiom of choice let's you create a set that has exactly one sock from each pair.
Doesn't this axiom seem plausible? Or in your terminology... Doesn't this axiom seem to be self-evident from objective reality?
It would seem unreasonable to assert that you could NOT create a new set that contains one element from each non-empty set in this collection.
You have a bunch of boxes lying on the floor. You should be able to pick one item from each box and put each one of these items into some new box. Shouldn't you?
However, this rather innocuous axiom has many bizarre and counter-intuitive consequences. The most famous is probably the Banach-Tarski paradox.
If you accept the Axiom of Choice, then you can logically deduce the Banach-Tarski theorem, which roughly asserts the following:
You can partition a 3-dimensional unit ball into finitely many pieces and using only rotations and translations, you can reassemble the pieces into two-balls each with the same volume as the original ball.
You can also show using a similar argument that you can cut a ball the size of a pea into finitely many pieces, rearrange these pieces, and end up with a ball the size of the sun.
To put it in a slightly humorous fashion...
If you believe that the axiom of choice is self-evident from objective reality, then you have to also believe that we can solve world hunger by cutting a single pea into pieces and rearranging the pieces to create a huge pea or as many peas as you want of the original size.
Maybe Jesus used the same method to multiply fish and bread!
;)
The fundamental question remains the same:
How do we choose our axioms?
It might be nice to think as the Greeks did that our axioms are self-evident truths that are abstracted directly from observing reality.
However, it seems more realistic and pragmatic to say that we choose the axioms we do, because they seem to work well. We try to choose axioms that seem to capture our intuitions about sets and numbers and the world in general. And then we can use the mathematical theory we've created from these axioms to model phenomena in the world such as bacteria growth and the trajectory of photons traveling through space.
Why do we use Zermelo Fraenkel Set Theory?
Because there is a social consensus amongst most mathematicians that it seems to do the job well.
Of course, one could argue that we should all have blind faith in the Zermelo Fraenkel axioms, because we have determined through a process of observing reality and abstracting that these axioms are indeed self-evident truths about objective reality.
However, I prefer to believe that axioms are chosen by social consensus based on our intuitions and our goals. And, hence I also believe that mathematical truths are indeed a social construct that somehow captures and coincides to some extent with our intuitions about the world.
Henrik.
[ Edited Wed Feb 25 2004, 01:22PM ]
--henrik Sun Feb 22 17:30:28 2004
sunny - Tue Feb 24 20:50:58 2004
| Doesn't this axiom seem plausible? Or in your terminology... Doesn't this axiom seem to be self-evident from objective reality? |
| -- henrik |
No. This axiom is not self-evident. It is a derived concept that appears to be taken out of its intended context through the process of treating it as an axiom. This is an example of a type of axiomatic fallacy, where a statement is supposed to be an axiom based on intuition or 'plausibility' as you stated. It may have a use within constructed human systems, but if this statement about sets is true then it is the lack of context in application that creates the paradox. It is missing a restriction that enables its proper application. This tells me directly that it is not an axiom.
I am not arguing for 'blind faith' as you stated. The truths that we hold self-evident are those that are directly observable. This is not 'faith', this is perception and observation. We do not contradict true knowledge by developing our science, we simply expand upon what is known to define the underlying details and intricacies that were not previously observable. Axioms, especially, must be scrutinized critically for coherence with reality. As soon as they lead to contradictions with reality it is our responsibility as scientists to take it upon ourselves to critically asses and dismantle false reasoning and false axioms.
Your view is a very subjectivist standpoint. Your arguments for this view as valid seem appropriately fuzzy and vague. Your arguments against an objective viewpoint are not valid as they assume that I must have 'blind faith' in my means of perception in order to perceive anything objectively. My means of perception, and yours, are valid. I do not have faith, or belief. I have observation, reason, and conclusion.
Do I have to 'believe' that I exist? No. This is axiomatic.
As stated by Rand: "Existence exists ó and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists."
You are right about mathematics being a game, but only within itself, pure mathematics, given the assumed validity of a set of arbitrary axioms. I can create new axioms that are true within the confines of a human constructed system; like a software system. Those axioms lead to a certain game-theoretic symbolic system that may or may not correspond to other systems. This does not invalidate the objectively true formulation of mathematics. It is simply an instance of application of the tools created by Mathematics, the science, to a human constructed system.
Mathematics currently holds many false assumptions, false axioms, and broken logic. The shear prevalence of the type of thinking that you are upholding as the pinnacle of mathematical 'formalism' is a dangerous sign that mathematics is slowly sliding toward the type of innaplicable societal trivialities and social constructions that you speak of. Mathematics that is not applied is never tested. If a physicist told you that their knowledge didn't need to ever be applied to be true you could be sure that he was a quack. I know full well that the majority of mathematicians would adamantly dissagree with me, but without application, without objectivity, mathematics is, as you stated, nothing more than a game. But with these important components, it is one of our most valuable sciences.
--sunny Tue Feb 24 20:50:58 2004
henrik - Wed Feb 25 10:01:17 2004
| Your view is a very subjectivist standpoint. |
| -- sunny |
I have been accused of that before. However, I prefer the notion of intersubjectivism. Knowledge does not depend solely on the subject, nor does it depend solely on the object. Knowledge arises as an interaction between objects and a community of subjects that interact with each other.
| My means of perception, and yours, are valid. I do not have faith, or belief. |
| -- sunny |
Well, don't forget that you do at least need faith that our means of perception are valid. If you don't have this belief, it is hard to move forward.
:)
| [The Axiom of Choice] is missing a restriction that enables its proper application. This tells me directly that it is not an axiom. |
| -- sunny |
In the sake of fairness, I will mention a few things that support what you are saying.
There is a minority of mathematicians, who take a similar stance as you do.
Some of them have tried to rebuild classical mathematical theories without ever using the Axiom of Choice or a weakened version of this axiom.
It is fairly surprising how much mathematics one can rebuild without this axiom.
However there are certain things that become out of reach without the full power of the axiom of choice.
Actually, ZF refers to Zermelo Fraenkel Set Theory with the Axiom of Choice and ZF- refers to the same theory without this axiom.
I find it interesting to compare the two theories, but I do not think that one is more 'valid' in any sense than the other is.
| Mathematics currently holds many false assumptions, false axioms, and broken logic. |
| -- sunny |
Broken logic? What is it about first order logic that seems broken to you? Do you have something concrete in mind?
Could you give me concrete examples of these 'many' false assumptions and false axioms of which you speak?
I know you seem to dislike the Axiom of Choice. Is there anything else in modern mathematical theory that seems to bother you?
| The shear prevalence of the type of thinking that you are upholding as the pinnacle of mathematical 'formalism' is a dangerous sign that mathematics is slowly sliding toward the type of inapplicable societal trivialities and social constructions that you speak of. |
| -- sunny |
I very strongly disagree with you. Progress in mathematics is not in one-to-one correspondence with how applicable a mathematical theory is. It sometimes takes decades or even centuries before a mathematical theory finds applications in science and engineering. There tends to be a considerable lag between theory and application.
A good example would be Non-Euclidean Geometry. This was invented in the first half of the nineteenth century. Physicists, philosophers and mathematicians accused Riemann of inventing a mathematical formalism that was useless. According to them, it was based on the flawed axiom of denying Euclid's fifth postulate, which directly contradicted our observations of physical reality. A century later, Non-Euclidean geometry was exactly the mathematical formalism that Einstein needed to capture the intuitions he had about physical reality and thus develop relativity theory.
I think you are biased into thinking that mathematics that leads to theories that don't seem to have an application or do not seem to reflect our current views about reality are not worthwhile pursuing.
| Mathematics that is not applied is never tested. If a physicist told you that their knowledge didn't need to ever be applied to be true you could be sure that he was a quack. |
| -- sunny |
I think you are confusing things by comparing mathematics and physics in that way.
Mathematical statements are valid based on axioms and logic alone.
Physics and other sciences require experiments to justify their statements.
If you use mathematical theory X, to describe a set of phenomena Z, and then you run some experiments to test if X makes good predictions about Z. You are NOT testing the mathematical theory itself. You are really testing how well you chose a mathematical formalism to represent phenomena Z.
| I know full well that the majority of mathematicians would adamantly disagree with me, but without application, without objectivity, mathematics is, as you stated, nothing more than a game. |
| -- sunny |
You are taking the same position as those physicists, philosophers and mathematicians trying to discredit the value of Riemann developing Non-Euclidean Geometry.
| [Mathematics] is one of our most valuable sciences. |
| -- sunny |
It is interesting that you mention this. I don't think that mathematics is a science. Or, at the very least, mathematics is not a science in the same way that physics and biology is a science.
I think that it would be more appropriate to say that mathematics is a language for science.
--henrik Wed Feb 25 10:01:17 2004
sunny - Wed Feb 25 21:36:56 2004
I think that you have mis-read me.
My 'biases' are quite the opposite or your statements. I know that mathematics is worth pursuing REGARDLESS of direct application BECAUSE it is rigorous and bases its axioms on existentially self-evident truths. It is the misuse of the concept of 'axiom' that is the prevalent example of 'broken logic' that I speak of. Show me a contradiction, cyclical truth, paradox, or indeterminant statement and I will show you how its logic is faulty and its application ignored.
Regarding social subjectivism:
Truth is not ever determined through a process of attrition. Group belief without validation is a self-destructive process that has precipitated every major human catastrophe and underlines human fault with an exclamation point screaming for a real answer.
Regarding the science of mathematics:
You, yourself stated that you considered mathematics a 'social' construction. I am simply pointing out why you have this view. Mathematics IS a game when you apply it as a game. It is a tool of science, when you apply its rules to science. It is a tool for determining objective validity of its own basis: within the scope of its application as a tool for abstracting from primary (perhaps contrived) axioms. It is a science when its axioms themselves are objective truths.
This same statement applies for any body of knowledge. This does not mean that it is subjective, or arbitrary. It means that it is dependent upon the context of its application.
Regarding 'faith': Try to prove something, anything, by presupposing 'faith' as an axiom as you have just done. I can prove that it leads to exactly what it is: a dead end. It is a false shortcut used when individuals choose not to think for themselves. Never close your mind by rescinding it to faith. It is irresponsible and self-condeming. Have confidence. Have pride. Have virtue. Don't discard the tough questions to the void.
We must dispense with the logic/anti-logic and get back to the question at hand:
Of critical importance to this discussion: What is an axiom?
You still have not answered my claim that the Poll is faulty, but you seem to have re-used some of the characteristics of my arguments: 'Object and Observer' That is exactly what I have in mind.
--sunny Wed Feb 25 21:36:56 2004
sunny - Wed Feb 25 21:41:08 2004
| Progress in mathematics is not in one-to- one correspondence with how applicable a mathematical theory is. It sometimes takes decades or even centuries before a mathematical theory finds applications in science and engineering. There tends to be a considerable lag between theory and application. |
| -- henrik |
This does not have to be the case. It is a failing of mathematics and the philosophy of mathematicians that this is true.
--sunny Wed Feb 25 21:41:08 2004
henrik - Thu Feb 26 11:07:55 2004
You make the following claims:
| I know that mathematics is worth pursuing REGARDLESS of direct application BECAUSE it is rigorous and bases its axioms on existentially self-evident truths. |
| -- sunny |
and...
| The shear prevalence of the type of thinking that you are upholding as the pinnacle of mathematical 'formalism' is a dangerous sign that mathematics is slowly sliding toward the type of innaplicable societal trivialities and social constructions that you speak of. |
| -- sunny |
Which one is it?
Are mathematical consequences like the Banach Tarski Paradox worth pursuing or not?
Are mathematical theories like Non-Euclidean Geometry and Non-Standard Analysis worth pursuing or not?
--henrik Thu Feb 26 11:07:55 2004
henrik - Thu Feb 26 11:21:13 2004
| Progress in mathematics is not in one-to- one correspondence with how applicable a mathematical theory is. It sometimes takes decades or even centuries before a mathematical theory finds applications in science and engineering. There tends to be a considerable lag between theory and application. |
| -- henrik |
| This does not have to be the case. It is a failing of mathematics and the philosophy of mathematicians that this is true. |
| -- sunny |
I agree that this doesn't need to be the case. However, there is no reason to blame this on mathematicians. It simply takes time for mathematical knowledge to be transmitted from the hands of the mathematician to the hands of the scientist and engineer. Of course, it is definitely worthwhile to try to streamline this process.
--henrik Thu Feb 26 11:21:13 2004
henrik - Thu Feb 26 11:49:17 2004
| Of critical importance to this discussion: What is an axiom? |
| -- sunny |
I couldn't agree more. I think that our fundamental disagreement boils down to the way view ëaxiomí.
You view the concept of axiom in the same way as the Ancient Greeks did as some sort of self-evident truth. A statement that must be true just by the mere nature of reality itself.
On the other hand, I have a much more general concept of what an axiom is. I view an axiom as a primitive truth that one accepts before starting the investigation of its consequences. However, I make no epistemological claim by saying that it is a 'self-evident' truth that we can somehow objectively abstract directly from reality.
| It is the misuse of the concept of 'axiom' that is the prevalent example of 'broken logic' that I speak of. Show me a contradiction, cyclical truth, paradox, or indeterminant statement and I will show you how its logic is faulty and its application ignored. |
| -- sunny |
What makes you think that mathematics based on ZF has contradictions?
Or do you have a problem with ZF mathematics being incomplete?
[ Edited Thu Feb 26 2004, 05:32PM ]
--henrik Thu Feb 26 11:49:17 2004
sunny - Tue Mar 2 19:20:35 2004
| You view the concept of axiom in the same way as the Ancient Greeks did as some sort of self-evident truth. A statement that must be true just by the mere nature of reality itself. |
| -- henrik |
No, you have the wrong impression of my perspective. VERY FEW 'axioms' are self evident. MOST 'axioms' are 'construct-primitive' truths that fit the rules of a given human construct. This is the differentiation between mathematics, the science, and mathematics as it may be applied. Objectively true axiom derived direclty from methaphysics => true axiom => the science of mathematics. On the other hand, simply representing the underlying assumptions and premises of some arbitrary constructed system => 'axiom-of-construct' => the application of mathematics to human systems. The application of formal rules do not require self-consistency or consistency with objective reality in the case of an 'axiom-of-construct' application of axiomatic systems. The construct itself could embody internal contradictions and direct conflicts with objective reality in its structure and underlying assumptions.
I can always 'construct' a set of rules that chooses arbitrarily the truth or falsity of basic premises. When these rules represent the characteristics of a human system, it is truth about this system AND ONLY this system to which a formal logic/symbolic system applies.
I can also determine from my direct observations those axioms that directly represent that which I observe to be true and real that could not have been otherwise. These axioms will lead me to conclusions in my logic/symbolic system that are also true and real that could not have been otherwise.
Regarding the dichotomy that you presume:
| Are mathematical consequences like the Banach Tarski Paradox worth pursuing or not? Are mathematical theories like Non-Euclidean Geometry and Non-Standard Analysis worth pursuing or not? |
| -- henrik |
Also regarding ZF and the Axiom of Choice:
There is no dichotomy. These are precisely the questions that I believe that I have an objective answer for and why this new answer is correct. I will provide this answer we meet.
--sunny Tue Mar 2 19:20:35 2004
charmed_quark - Fri Mar 5 9:51:21 2004
| Are you implying that a process of quantification in measurement is not a process of observing objective reality? Mathematics itself is something derived directly from fundamental perceptual observations of objective reality. |
| -- sunny |
That is exactly what I am implying. I will go further by saying that mathematics is an entirely human constructed concept. Mathematics is not derived from observation, but is derived from axioms (see any Introduction to Real Analysis textbook). These axioms amount to throwing up our hands and saying 'that's just the way it is'
I will reemphasize that math is the most useful and successful model humans have invented, but that is not sufficient to conclude that it reflects reality in any fundamental way.
If you would kindly list a book, not by Rand, that gives an idea of this objectivist philosophy, I will look into it.
--charmed_quark Fri Mar 5 9:51:21 2004
sunny - Sun Mar 14 7:11:20 2004
Have you read any of the book listings that I have provided thus far?
A very well written book on the ideas of objectivism: 'Objectivism', the Philosophy of Ayn Rand by Leonard Peikoff. It is the best non-Rand source that I have found and talks to the issue at hand. That is, the problem of subjectivism.
Regarding the intent of Subjectivism and Objectivism, consider this:
Subjectivism would hold that all choices are equally valid. Therefore, every choice is the best choice, and every choice is the most correct choice. The corollary to this would be: no choice is a 'better' choice, and no choice is a 'more' correct choice
According to Objectivism this must be false.
Objectivism would hold that: there is an objectively correct choice and this choice is the best choice.
The reality-consistent axiom is the objectively correct choice and the best choice (of all possible axioms) we are able to make given our present knowledge of reality. All axioms are not equally valid. Ask a rational logicician. Henrik?
Mathematics is objective because it is tied solidly to reality, not in spite of the fact that it isn't.
[ Edited Sun Mar 14 2004, 09:02PM ]
--sunny Sun Mar 14 7:11:20 2004
