Philosophy Discussion Forums

MazerRackhem wrote:I find it so interesting that you dismiss anything I say because of my credentials but are quick to grasp at anything smacking of legitimacy in the theories you cling to.

It seems you have let your prejudices against creationism rule your opinion about the theory. Those scientists aren't creationists themselves, so the prejudice doesn't even apply. But some of them are involved with alternative medicine, so then you can let your prejudice against alternative medicine rule your opinion about the theory? You underestimate the power of logic of rewriting, this is most likely THE theory of everything because one cannot get more fundamental than 0. While many scientists are looking for ever more flimsy fundamental particles, the existence of which is ever more abstractly theoretically confirmed, it means empiricial research is pointing towards nothing as the fundament to the universe, which is accurately described with 0.

Hi Mazer

Before this thread gets too far off topic.... I would be interested in any response you might have to my post #28.

A_Seagull wrote: But why this concern with 'completeness'?

I should perhaps have been a bit clearer in my definitions. By completeness here I am referring to the completeness of the Reals. That is, using Dedekind Cuts we can easily prove that there are no gaps in the real numbers. There of course remains the complex numbers which are required to solve for the roots of certain polynomials. I meant completeness only in this fashion. I think that the continuity of the Reals is a fairly important property to establish. Though one can argue that the definition of a Real Number via a Dedekind cut is really "no better off" than merely accepting the existence of the Reals axiomatically, in my opinion the ability to prove certain qualities (such as completeness) using an abstract logical framework provides a more satisfying foundation. I can only speak from personal experience, but when I first encountered Dedekind Cuts in Real Analysis I felt I had achieved a much deeper understanding of the concept of "number" than I had previously held. I don't think mathematics requires the Dedekind Cut definition, but I do think it is enriched by it.

A_Seagull wrote: If I have a real number - take 1.5 for example, that defines it - at least within the general framework of mathematics. I do not need any further definition.

Well that isn't precisely true. When you use the symbol 1.5 you have an idea in your mind of what that entity is. You may think of one and one half for instance, which is built upon the dual definitions of "one" and the concept of "half." The question of course is what is "one" or "1?"We may choose to define the object 1 as the base unit object and to define addition such that 1+1=2 where 2 is defined as the object 1+1. This agrees well with our experience in real world enumeration because we can easily refer each integer back to a group of objects which it enumerates and verify our arithmetic with physical stand ins, it does however, (in my humble opinion) lack a degree of 'elegance.' We also become a bit befuddled in attempting to tie the irrationals, and certainly the transcendental numbers, down via our experience with real world objects.

By using an abstract notion of number not tied to our experience of enumeration (I hope I have convinced you that we must have some concept of what 1.5 is which extends beyond the printing of symbols to paper) we are able to view addition, subtraction, multiplication, enumeration, etc. as a logical process easily defined and applied to independent abstract entities rather than having to define a number in terms of its relation to a base unit of enumeration. Thus number 1.5 is indeed "one and one half" but this is a property which evolves out of the definition of number as a cut in the continuum.

Here I must admit I feel I am not fully conveying the essence of the idea which is involved. Perhaps the best that I can say is that, personally, I view that Dedekind cut definition as conveying a much more satisfying sense of what a number is. That is, that a number is not an enumeration of a group of objects, but rather a sort of sharp point in the continuum. I think this view of number lends itself very well to Real Analysis and especially to the concept of probability where we can view numbers as stand in tokens for statistical events. For instance I think the value of a probabilistic retrieval of an integer from the line at a given draw is more intuitively understood when viewing number from this perspective.

A_Seagull wrote: Bertrand Russell's 2 volume 'proof' of '1+1=2' seems to me to be a total absurdity.

I more or less agree. While his motives were sound I doubt too much good could have come from such a reduction in the best of circumstances. Thanks to Godel we now know that the ultimate good sought after (namely the ability to create a writeable system in which all of mathematics could be drawn up algorithmically) is impossible. Had this not been the case the effort may have produced a system for radically advancing mathematics. Since the overall program was proven an impossibility however I don't think anything useful can ever be obtained from the effort.

A_Seagull wrote: My understanding is that Godels work followed from Hilbert's suggestion that mathematics might be finitely describable, consistent and complete. So, Godel proved that this could not be so.

Yes, Godels theorems are often seen as a negative solution to Hilbert's second problem.

A_Seagull wrote: Clearly the axioms should be finitely describable and if inconsistency is found then this is not a 'good' thing. But there is no necessity for completeness of the system. The theorems generated from the axioms can happily continue wihout check or covering every possibility.

The problem is of course in the finding of the inconsistency. Godel's second theorem demonstrates that no such system can be proven consistent. Also while it is true that mathematics does not need to prove every possible theorem to be a useful and worthwhile endeavor it is somewhat unsettling that there must exist theorems in any "theory" (using Godel's language) of arithmetic which are forever beyond the bounds of proof. Would we be satisfied with a mathematical formalism which forever precluded the answering of the Riemann Hypothesis? I doubt it.

Godel's work speaks to our inability to "explain mathematics to a computer" as it were. Some have interpreted this as having some bearing on the ability of organic humans to transcend the capability of digital processes and used it as a "proof" that we are more than biological machines. I think this is overstepping the theorems bounds. It does however engender certain questions regarding the foundations of mathematics and our ability to comprehend the absolute roots of this logical system.

As another passing note: Have you ever heard of or read the book Godel, Escher, Bach. An Eternal Golden Braid ? It has been a while since I first picked it up and I should go back and read it again now that I actually know a bit more about what the author is talking about. If you're interested in these sorts of questions you may find it an enjoyable read.

A question for the participants on this thread:

Imaginary numbers (i.e., numbers which incorporate √-1 as a factor) have very "real" application in modeling various engineering scenarios, especially at the quantum scale. Yet I am not familiar with anyone who has claimed that √-1 is anything more than an abstraction. Am I wrong about that? And if indeed √-1 doesn't correspond to any real-world "concrete" in any manner, how would any presumed foundation for mathematics accommodate it?

A Poster He or I wrote:A question for the participants on this thread:

Imaginary numbers (i.e., numbers which incorporate √-1 as a factor) have very "real" application in modeling various engineering scenarios, especially at the quantum scale. Yet I am not familiar with anyone who has claimed that √-1 is anything more than an abstraction. Am I wrong about that? And if indeed √-1 doesn't correspond to any real-world "concrete" in any manner, how would any presumed foundation for mathematics accommodate it?

IMO:

All of pure mathematics is an abstraction. It has no correlation with the real world (or anything concrete) except through a mapping between a mathematical concept and a real world concept. This mapping is necessarily inductive - (as opposed to the deductive inferences that are the essence of pure mathematics.)

So even the correlation of integers with building blocks, that children play with, is a mapping between the two. It is not an essential part of pure mathematics - even though the correlation between the two may seem 'obvious'.

So then it is no problem to also view the correlation between 'imaginary' numbers and aspects of the real world (such as ac electricity) as just another inductive mapping. It is an inductive mapping as there is no deductive rationale for mapping imaginary numbers to ac electricity. Just as there is no deductive rationale for mapping calculus to moving objects or integers to building blocks.

The implications for the foundation of mathematics is that the foundations of pure mathematics can be as abstract as mathematics itself.

A Poster He or I wrote: Imaginary numbers (i.e., numbers which incorporate √-1 as a factor) have very "real" application in modeling various engineering scenarios, especially at the quantum scale. Yet I am not familiar with anyone who has claimed that √-1 is anything more than an abstraction. Am I wrong about that? And if indeed √-1 doesn't correspond to any real-world "concrete" in any manner, how would any presumed foundation for mathematics accommodate it?

No foundation of mathematics need contain within it a full description of all of the entities which arise from it. For instance, using a set theoretic basis for algebra leads to the definition of a Dedekind Cut which leads to a description of the reals, the foundational system need not contain an a priori description of the reals if they can be constructed from the base system itself. Likewise, using any system in which we build up the real numbers, we may further define the complex numbers in terms of the solutions to certain polynomials. That is, having established (a rather simple thing to do) that the polynomial x²=-1 has no solution in the reals we can assign the value of its solution the symbol "i." From this base unit the remainder of complex analysis may be built up.

As you have noted, while the complex numbers in and of themselves do not always correspond to real world entities directly they have proven vital in the solution of may practical problems, the Heat Flow or Wave Equation differential equation problems are two common examples encountered by undergraduate Physics, Math, Engineering, and Chemistry students. While the complex numbers do stretch the mind somewhat in our conception of what precisely they mean in correspondence with the physical world so do other elements of the reals. The Transcendental numbers for instance are difficult to put into a true correspondence with physical reality since they have neither a fractional representation nor are the root to any rational polynomial equation. Complex numbers by contrast (which can also be Transcendental) sometimes correspond to specific real world phenomena. In my own research for instance, the scattering of electromagnetic radiation is recorded using a carrier wave modulated signal which is decoupled at the receiver and the input is recorded as a complex numerical solution to a simple wave equation demodulation formula. Both the real and the imaginary parts of the recorded number have a physical significance when considering the received waveform.

Complex numbers and complex algebras can also be introduced in a "group theory" format in which each complex number is treated as an element of the set of all ordered pairs with appropriate algebraic rules and a set metric (most commonly the Euclidean metric). So that the number 7+2i is written as (7,2) or as the vector <7,2>. The "i" symbol is then seen as a sort of placeholder to separate out the components of the vector. With appropriately defined laws of addition, multiplication, etc. the entirety of complex analysis can be accomplished without the need to define any imaginary quantities. Thus use of the symbol "i" and the term "imaginary" is most often continued out of convenience and for historical reasons rather than out of necessity. When viewing complex analysis as merely the application of mathematics on the set of ordered pairs it is easy to view the "complex" number 7+2i as actually more closely tied to the physical world than is the odd Trancendental number.

In my personal opinion, although complex analysis can be constructed without the need to discuss the imaginary base unit, doing so imparts a better intuitive understand of the underlying mathematics and yields a more fruitful grasp of the growth of the reach and breadth of mathematics over the years. The move from integer enumeration to include the negative integers, to then including the rationals, followed by the discovery and acceptance of the irrationals, the discovery of the transcendentals, and finally, culmination in the inclusion of the complex numbers which has yielded up the Fundamental Theorem of Algebra and the proof that the set of numbers is at last "complete" in that all polynomials are now solvable for every possible root within the current set of numbers.

A_Seagull said,

The implications for the foundation of mathematics is that the foundations of pure mathematics can be as abstract as mathematics itself.

I agree with every word verbatim of your post # 41.

Mazer said,

...although complex analysis can be constructed without the need to discuss the imaginary base unit, doing so imparts a better intuitive understand of the underlying mathematics and yields a more fruitful grasp of the growth of the reach and breadth of mathematics over the years.

I have no problem with this since I don't see it as incompatible with A_Seagull's position as stated in post # 41. It seems pretty clear from your posts that you appreciate the foundation in terms of continuity with its derived particulars while at the same time avoiding Russell & Hilbert's hubris that the foundation itself cannot be an abstraction.

MazerRackhem wrote: (Nested quote removed.)


I should perhaps have been a bit clearer in my definitions. By completeness here I am referring to the completeness of the Reals. That is, using Dedekind Cuts we can easily prove that there are no gaps in the real numbers. There of course remains the complex numbers which are required to solve for the roots of certain polynomials. I meant completeness only in this fashion. I think that the continuity of the Reals is a fairly important property to establish. Though one can argue that the definition of a Real Number via a Dedekind cut is really "no better off" than merely accepting the existence of the Reals axiomatically, in my opinion the ability to prove certain qualities (such as completeness) using an abstract logical framework provides a more satisfying foundation. I can only speak from personal experience, but when I first encountered Dedekind Cuts in Real Analysis I felt I had achieved a much deeper understanding of the concept of "number" than I had previously held. I don't think mathematics requires the Dedekind Cut definition, but I do think it is enriched by it.


(Nested quote removed.)

Well that isn't precisely true. When you use the symbol 1.5 you have an idea in your mind of what that entity is. You may think of one and one half for instance, which is built upon the dual definitions of "one" and the concept of "half." The question of course is what is "one" or "1?"We may choose to define the object 1 as the base unit object and to define addition such that 1+1=2 where 2 is defined as the object 1+1. This agrees well with our experience in real world enumeration because we can easily refer each integer back to a group of objects which it enumerates and verify our arithmetic with physical stand ins, it does however, (in my humble opinion) lack a degree of 'elegance.' We also become a bit befuddled in attempting to tie the irrationals, and certainly the transcendental numbers, down via our experience with real world objects.

By using an abstract notion of number not tied to our experience of enumeration (I hope I have convinced you that we must have some concept of what 1.5 is which extends beyond the printing of symbols to paper) we are able to view addition, subtraction, multiplication, enumeration, etc. as a logical process easily defined and applied to independent abstract entities rather than having to define a number in terms of its relation to a base unit of enumeration. Thus number 1.5 is indeed "one and one half" but this is a property which evolves out of the definition of number as a cut in the continuum.

Here I must admit I feel I am not fully conveying the essence of the idea which is involved. Perhaps the best that I can say is that, personally, I view that Dedekind cut definition as conveying a much more satisfying sense of what a number is. That is, that a number is not an enumeration of a group of objects, but rather a sort of sharp point in the continuum. I think this view of number lends itself very well to Real Analysis and especially to the concept of probability where we can view numbers as stand in tokens for statistical events. For instance I think the value of a probabilistic retrieval of an integer from the line at a given draw is more intuitively understood when viewing number from this perspective.




(Nested quote removed.)


I more or less agree. While his motives were sound I doubt too much good could have come from such a reduction in the best of circumstances. Thanks to Godel we now know that the ultimate good sought after (namely the ability to create a writeable system in which all of mathematics could be drawn up algorithmically) is impossible. Had this not been the case the effort may have produced a system for radically advancing mathematics. Since the overall program was proven an impossibility however I don't think anything useful can ever be obtained from the effort.


(Nested quote removed.)


Yes, Godels theorems are often seen as a negative solution to Hilbert's second problem.


(Nested quote removed.)


The problem is of course in the finding of the inconsistency. Godel's second theorem demonstrates that no such system can be proven consistent. Also while it is true that mathematics does not need to prove every possible theorem to be a useful and worthwhile endeavor it is somewhat unsettling that there must exist theorems in any "theory" (using Godel's language) of arithmetic which are forever beyond the bounds of proof. Would we be satisfied with a mathematical formalism which forever precluded the answering of the Riemann Hypothesis? I doubt it.

Godel's work speaks to our inability to "explain mathematics to a computer" as it were. Some have interpreted this as having some bearing on the ability of organic humans to transcend the capability of digital processes and used it as a "proof" that we are more than biological machines. I think this is overstepping the theorems bounds. It does however engender certain questions regarding the foundations of mathematics and our ability to comprehend the absolute roots of this logical system.

As another passing note: Have you ever heard of or read the book Godel, Escher, Bach. An Eternal Golden Braid ? It has been a while since I first picked it up and I should go back and read it again now that I actually know a bit more about what the author is talking about. If you're interested in these sorts of questions you may find it an enjoyable read.

I am not convinced that mathematics needs to be tied to the real world. The whole idea of pure mathematics (as opposed to applied mathematics) is that it is not tied to the real world. If mathematics were to be tied to the real world one would need to be clear which bits need to be tied and which do not.

I have flicked through 'Godel, Escher and Bach' on occasions. It is a little whimsical for my taste. However I enjoyed 'Mind Tools' by R. Rucker. In it he postulates that pure mathematics can be modelled by a theorem generating machine. The machine is created purely from the axioms of the system. It takes the axioms of symbols and operations and generates theorems. These theorems are then necessarily 'true' within the system. And there would seem to be no reason why a machine (albeit hypothetical) could not generate all the theorems of mathematics.

The only requirement for consitency is that the axioms of the system allow for a unique machine to be 'created' which incorporates all of the axioms.

It is a moot point whether it could generate Godel's Incompleteness theorem. But I see no reason why not if the axioms are selected appropriately.

So if mathematics can be generated from the simple axioms of mathematics there is no need for the convoluted ideas about sets to be included in the axioms of the system. (As referred to in Post #1 of this thread). Nor is there any need for Peano's axioms.

As suggested in post #41 above , the relationship between pure mathematics and the real world can be achieved through an inductive mapping.

Personally I would have no problem if the Riemann hypothesis was never proved one way or another, nor the Goldbach conjecture for that matter. It may simply be the nature of the mathematical universe. In the same way that it would seem impossible to look beyond the event horizon of a black hole. It's something I just have to accept as part of the nature of the real world.

Animals. We're all animals. Before mathematics there was no way to count the number of toes on our feet. Then math came along and revolutionized this. Since then we animals have put a number on every thing. And after all it belongs to that class of intellectual tools that over complicate ordinary life - so much so that an academic system may exist. This has been the aim and goal of math since the lazy Greeks had nothing better to do with their times. And it is an excuse today for what science and pure number men call "work".

Meanwhile thousands suffer from starvation and though the calculations are simple to figure out what number of people and x amont of rice could help third world countries, the physical action of giving aid, or even improving the standards in a persons local area go unproved. Can it be done? According to the numbers, yes. As far as everyone is concerned, apparently not. So math is obviously a simple model. Science is little more than fishing in a pond for frogs. For shame academia. I recall a women once asked a NASA man to justify all the money granted to space exploration in light of all the existing social may lay. His response was that the "romance" of space exploration was an "intrinsic" desire of man and always had been. He made no comment on the conditions of the poor. The poor were not expressive of an intrinsic, human desire to "touch the stars".

Science would prove itself detrimental to the world, in the field of animal studies and would be forced to abandon all further pursuits. For shame on science for taking the animal out of the human. Prudes.

A_Seagull wrote:
Personally I would have no problem if the Riemann hypothesis was never proved one way or another,

Quoted from the previously provided reference in this thread: (my understanding of this is; something, something, something, therefore something, which means nilpotence theory hypothetically proves the Riemann hypothesis. My interest is mainly that the theory further proves freedom is real and the validity of the logic of creation, which validates subjectivity.)

document :http://www.naturescode.org.uk/userfiles ... S2008a.pdf

Riemann Hypothesis
Bernard M. Diaz, Peter Marcer, Peter Rowlands

Abstract
A novel approach to a proof of the Riemann Hypothesis (that all the zeros of the Zeta
function lie on the line x= ½) is presented. It is based on the universal nilpotent
computational rewrite system (NUCRS), derived in the World Scientific book Zero to
Infinity, from a single nilpotent Dirac operator (Rowlands, 2007), establishing an
entirely novel semantic computational foundation simultaneously for both mathematics
and quantum physics. Tangible evidence is that the Zeta function is known to represent
a quantum system and that the criterion of nilpotence corresponds to(a) Pauli exclusion
with unique fermion states spin ½ and (b) an infinite rewrite alphabet that also
corresponds to the infinite roots of –1, of which the nilpotent generalization of Dirac’s
famous quantum mechanical equation is the universal computational order code.
Keywords:the Riemann Hypothesis, the nilpotent computational rewrite system,
quantum mechanics, the nilpotent Dirac operator, the infinite roots of –1.

A_Seagull wrote:In it he postulates that pure mathematics can be modelled by a theorem generating machine. The machine is created purely from the axioms of the system. It takes the axioms of symbols and operations and generates theorems. These theorems are then necessarily 'true' within the system. And there would seem to be no reason why a machine (albeit hypothetical) could not generate all the theorems of mathematics.

This is precisely what Godel proved to be impossible. No such machine can ever prove all of the theorems of mathematics. For any given machine there must exist true theorems in the language of the machines axioms and operations which it can never prove. This is probably the most easily grasped explanation of the Incompleteness Theorems. The machine you describe is precisely what the Logicist mathematicians were hoping to achieve by their reductions, Godel showed that all such efforts have a very real limit and that no machine can be built either in reality or hypothetically. This is why I was saying that the Incompleteness theorems have interesting things to say about the foundations of mathematics.

There is fundamentally no cause or explanation, justification or reason, that "two" should follow "one". I've heard that the number line is "well ordered". I don't think I have ever heard it being "necessarily ordered" or "ordered well". There is no order to the numebr line apart from the notion that "large is bigger than small". And that means nothing in terms of a necessary function of "putting things in order". I have a collection of acorns on a window sill ordered from smallest to largest. It fills the sill from end to end and I often wish I had a larger window and more acorns. But what makes me happiest about this collection is that no one is allowed to shuffle them around. When I watch squirrels collect acorns I assume they are doing it for the exact same reasons.

MazerRackhem wrote: (Nested quote removed.)


This is precisely what Godel proved to be impossible. No such machine can ever prove all of the theorems of mathematics. For any given machine there must exist true theorems in the language of the machines axioms and operations which it can never prove. This is probably the most easily grasped explanation of the Incompleteness Theorems. The machine you describe is precisely what the Logicist mathematicians were hoping to achieve by their reductions, Godel showed that all such efforts have a very real limit and that no machine can be built either in reality or hypothetically. This is why I was saying that the Incompleteness theorems have interesting things to say about the foundations of mathematics.

I think you misunderstand me. The machine would not produce all the thorems of 'mathematics' it would merely produce all the theorems that it was capable of producing. If, from the standpoint outside of the system, it appeared to be 'gaps' in the theorems then this would not be a problem. Godel's thorem only showed that no system of mathematics could completely fill all of mathematical space; ie for every possible statement 'S' no system would either generate 'S' or 'not-S'. But this does not preclude a sytem that does not produce either 'S' or 'not-S'. Yet such a system could generate theorems that relate to all of the known theorems of mathematics.

A_Seagull wrote: Godel's thorem only showed that no system of mathematics could completely fill all of mathematical space; ie for every possible statement 'S' no system would either generate 'S' or 'not-S'. But this does not preclude a sytem that does not produce either 'S' or 'not-S'. Yet such a system could generate theorems that relate to all of the known theorems of mathematics.

further quotation from a reference previously provided in the thread:
document: http://www.naturescode.org.uk/userfiles ... wlands.pdf

" By rejecting the ‘loaded information’ that the
integers represent, and by basing our mathematics on an immediate zero totality, we
believe that we are able to produce a mathematical structure that has the potential of
avoiding incompleteness in the Gödel sense. (Conventional approaches, based on the
primacy of the number system, have necessarily led to the discovery that a more
primitive structure cannot be recovered than the one initially assumed.) "

Which means according to me, that with nilpotency theory a universe (or mathematical structure) consisting of just nothing or zero, has equal priority over a sophisticated universe consisting of matter + anti-matter, because the totality of both configurations of the universe is equally nothing. There is then no incompleteness of scope over possible configurations, because non zero totality configurations correctly defines impossibility. A mathematical structure which does not total nothing, is on it's own an incomplete and impossible mathematical structure.

A_Seagull wrote: I think you misunderstand me. The machine would not produce all the thorems of 'mathematics' it would merely produce all the theorems that it was capable of producing.

Fair enough. I think we're talking past each other a bit with our ideas of completeness. A machine could theoretically devised which would prove every theorem which has thus far been proven (in the hypothetical of course). I do think it says something quite interesting about the foundations of mathematics though, that no such machine can ever prove all true statements in the system it is set up for.