Wolfram's View on Mathematics: Are There Limits to Formal Systems?

[Kherubin]

I would be much obliged if you could take a look at the video provided below:

http://www.closertotruth.com/video-profile/Is-Mathematics-Invented-or-Discovered-Stephen-Wolfram-/1384

On a couple of pages you can give all the axioms commonly in use in Mathematics. They are quite simple, but they are what you grow Mathematics from. The question is then, are those the only possible axioms?

I certainly wouldn't mind a more general discussion on the Platonic/Invented nature of Mathematics. However, I think that the PF Staff & Moderators would probably prefer, justifiably, if we kept that discussion to a minimum, because it is debated so often and far better elsewhere. Instead, shall we try to keep the discussion centered on the implications for Mathematics of Wolfram's views specifically.


In the way he delivers his answer, he seems to be suggesting that this more general view of mathematics does away with Platonism. Is this true? Surely it just passes the buck one step on. This wider sphere of formal systems could equally be Platonic too.

He states that many (perhaps all) of these formal systems are internally self-consistent. How can this be the case if the method he is using is simply the re-organization of mathematical symbols?

If its possible to produce new, entirely self-consistent axiomatic systems in this way, is there actually any limit to the number of said systems that can exist? Or are they innumerable?


Thanks for your input,
Kherubin

 

[SW VandeCarr]

I would be much obliged if you could take a look at the video provided below:

http://www.closertotruth.com/video-profile/Is-Mathematics-Invented-or-Discovered-Stephen-Wolfram-/1384 In the way he delivers his answer, he seems to be suggesting that this more general view of mathematics does away with Platonism. Is this true? Surely it just passes the buck one step on. This wider sphere of formal systems could equally be Platonic too.

He states that many (perhaps all) of these formal systems are internally self-consistent. How can this be the case if the method he is using is simply the re-organization of mathematical symbols?

If its possible to produce new, entirely self-consistent axiomatic systems in this way, is there actually any limit to the number of said systems that can exist? Or are they innumerable?Thanks for your input,
Kherubin

I was unable to connect to your link x3, but from your question, I would say there is no limit to the number of formal systems that can exist. A formal system doesn't have to mean anything, in the sense that it doesn't need an interpretation. It can just be a set of symbols together rules for symbol manipulation. Any string which can be so constructed is called a theorem of the system.

Usually such a system has:

1) A finite set of symbols.
2) A rule for a well formed formula; that is, a proper syntax for strings.
3) A set of initial formulas called axioms.
4) A set of rules for deriving theorems from axioms, called rules of inference.

In addition, formal systems may incorporate standard proof routines as well as defined and undefined terms.

EDIT: I finally was able to access your link. I'm fairly familiar with Wolfram's ideas. He published a huge tome about a decade ago exploring the possibilities of an alternative formal system. Essentially I agree with his view is that human mathematics is an historical artifact which has been very successful in modelling certain kinds of problems. However what we call mathematics contains several formal systems. We have Euclid's axioms for plane geometry. He really originated this idea. Since then, we have axioms for arithmetic, vector algebra and other areas of mathematics. Finally, by describing mathematical problems in terms of sets, we have the axioms of various set theories of which the Zermelo-Fraenkel axioms plus the Axiom of Choice (ZFC) is the current standard. But the point is that these axiomatic systems came later. Humans have been working with arithmetic, algebraic and geometric concepts long before they were formalized under sets of axioms.

 

[Kherubin]

Thank you for your reply.

I was unable to connect to your link

If anyone else has this problem, please:

1) Go to www.closertotruth.com
2) Search for Wolfram
3) The video is filed under, 'Is Mathematics Invented or Discovered'?

A formal system doesn't have to mean anything, in the sense that it doesn't need an interpretation. It can just be a set of symbols together rules for symbol manipulation.

Is this why all such systems are internally self-consistent, even though, in the case of Wolfram, they have been arrived at by the random rearrangement of symbols?

However what we call mathematics contains several formal systems. We have Euclid's axioms for plane geometry. He really originated this idea.

So, in this sense, the use of formal systems itself is a historical artifact? Is it possible to conceive of 'Mathematicses' which do not depend on this kind of structure? Would they be as rich as our Mathematics?

Thank you for your time and information,
Kherubin

 

[SW VandeCarr]

Thank you for your reply

Is this why all such systems are internally self-consistent, even though, in the case of Wolfram, they have been arrived at by the random rearrangement of symbols?

In a formal system, individual symbols are not random. So 2 -2 + 0 = is not a well formed formula (WFF) . So there needs to be rules for creating WFFs.

So, in this sense, the use of formal systems itself is a historical artifact?

Well, that's a good question. Aristotle explored syllogisms, but the study of formal systems as such is a relatively recent phenomenon.

Is it possible to conceive of 'Mathematicses' which do not depend on this kind of structure? Would they be as rich as our Mathematics?

Do you mean, something other than a formal system? There are any number of applied procedural applications which are "naive". However as a theoretical discipline, formal mathematics is currently based on the ZFC axioms. although not every true statement can be proven, nor can the consistency of the ZFC axiomatic foundation be established.

On the other hand, formal systems of limited scope can be developed which are known to be consistent and where every true statement can be proven.

 

[Kherubin]

In a formal system, individual symbols are not random. So 2 -2 + 0 = is not a well formed formula (WFF) . So there needs to be rules for creating WFFs.

So, when Wolfram talks of this Library of possible formal systems, each 'Formal System' actually refers to a set of symbols together with rules for symbol manipulation AND rules for creating WFFs?

There are any number of applied procedural applications which are "naive".

Could you please explain the notion of 'naive' in this context?

Do you mean, something other than a formal system?

Exactly! Is it possible to conceive of a self-consistent and useful 'Mathematics' not based upon the structure of a formal system?

 

[SW VandeCarr]

So, when Wolfram talks of this Library of possible formal systems, each 'Formal System' actually refers to a set of symbols together with rules for symbol manipulation AND rules for creating WFFs?

Yes, but the WFF requirement is a constraint. You can't order symbols any way you want. You must follow rules of the particular system. So 2 + (-2) = 0 is a WFF, but = ) 0 ( - 2 2 is not.

Would you please explain the notion of 'naive' in this context?

It simply means a set of procedural rules that lack some of the characteristics of a formal system listed in my first post. For example, naive set theory is not based on a formal set of axioms but relies instead on a few definitions and a simplified set of rules.

Exactly! Is it possible to conceive of a self-consistent and useful mathematics' not based upon the structure of a formal system?

Yes,provided it is limited in scope. The Egyptians built the pyramids without recourse to a proper formal system of mathematics, but they were able to do the necessary calculations aided by trial and error. The fact is, engineers and even physicists don't necessarily worry about formal proofs. They want models that predict experimental results.

In addition, Euclid's system for plane geometry is considered internally self-consistent and an obviously useful formal system, but there are alternative geometries where some of the axioms are changed or deleted, The classic example is the parallel postulate (axiom) which you should look up.

 

[Kherubin]

He published a huge tome about a decade ago exploring the possibilities of an alternative formal system.

I take it this was his New Kind of Science (NKS)? Did you read it? Was it any good?

Yes, but the WFF requirement is a constraint. You can't order symbols any way you want. You must follow rules of the particular system. So 2 + (-2) = 0 is a WFF, but = ) 0 ( - 2 2 is not.

So when Wolfram talks of "enumerating all possible such sequences of symbols" he is, in reality, referring only to those systems with the WFF constraints in place? Is this implicit or explicit? In other words, am I correct in assuming that with these WFF constraints, a system would not 'find itself' on the list of "all possible such sequences of symbols"?


Also, in taking the discussion back to my initial query, does Wolfram's position in any way alter the potential for 'Mathematics' to be 'Platonic'? Is this not just 'spreading the net' a little wider? Couldn't all these other formal systems also be 'Platonic' in nature?

Lastly, and I assure you I have had a look at the online description of these areas, but my brain is too small and not sufficiently mathematically adept to understand. Could you please elucidate the subtle difference between an axiomatic system and a formal system? Am I right in thinking that all axiomatic systems are formal systems, but the reverse is not necessarily true?

 

[SW VandeCarr]

I take it this was his New Kind of Science (NKS)? Did you read it? Was it any good?

Yes. It's not the kind of book you just read. It's more of a reference for those who want to explore the possibilities of these cellular automata. I'm not qualified to judge if it's any good.

So when Wolfram talks of "enumerating all possible such sequences of symbols" he is, in reality, referring only to those systems with the WFF constraints in place? Is this implicit or explicit? In other words, am I correct in assuming that with these WFF constraints, a system would not 'find itself' on the list of "all possible such sequences of symbols"?

Take natural language with 26 letters and some punctuation. You can write meaningful statements in many languages with these, but in every case there are rules for word formation and syntax. The same is true for formal systems. While there are just so many natural languages that exist or have existed, many more are no doubt possible.

Also, in taking the discussion back to my initial query, does Wolfram's position in any way alter the potential for 'Mathematics' to be 'Platonic'? Is this not just 'spreading the net' a little wider? Couldn't all these other formal systems also be 'Platonic' in nature?

This is a different question entirely. Your question is about formal systems. If you want discuss Plato's philosophy, you should start a new thread. In my opinion, Wolfram does not appear to be a Platonist.

Lastly, and I assure you I have had a look at the online description of these areas, but my brain is too small and not sufficiently mathematically adept to understand. Could you please elucidate the subtle difference between an axiomatic system and a formal system? Am I right in thinking that all axiomatic systems are formal systems, but the reverse is not necessarily true?

A formal system is an axiomatic system and an axiomatic system is a formal system as far as I know.

 

[Kherubin]

You can write meaningful statements in many languages with these, but in every case there are rules for word formation and syntax. The same is true for formal systems.

I was simply struck by his choice of language. Enumerating "all possible such sequences of symbols" sounds like a fairly arbitrary, random reorganizational process to me, but I now assume that what he has in mind is inclusive of the "rules for word formation and syntax" that you speak of.

This is a different question entirely. Your question is about formal systems. If you want discuss Plato's philosophy, you should start a new thread. In my opinion, Wolfram does not appear to be a Platonist.

Well, it is one of the reasons I started the thread in the first place, and the video is called 'Is Mathematics Invented or Discovered?'. I didn't want to start a brand new thread on Platonism, because it has been discussed fairly extensively elsewhere and I was most interested in the implications of Wolfram's views specifically.

I certainly agree with you that "Wolfram does not appear to be a Platonist". I simply do not see how his realization and enumeration of formal systems other than those commonly in use refutes the ideas of Platonism. Surely these 'new' formal systems can be just as 'real' and 'Platonic' in nature as those we have already discovered?

Thank you again,
Kherubin