Twistar wrote:Cool it looks like we mostly agree then.

Yes - I think so. By analogy - there is no question over the general process of evolution; but still some room for maneuver over the details...

So here is I think the crux of the disagreement and I think it is mostly our subjective perceptions. You say mathematics aims for "an impossible target." I'm not exactly sure what you mean by this but I'll take a stab at it.

I think that you think:

1) mathematics purports to convey absolute meaning in some way.

2) Due to limitations of linguistics this is impossible

3) therefore math is doing something that it shouldn't be doing and

4) thus you propose mathematics should instead aim to be "fully consistent with the limitations of language. If mathematics used language in the most efficient way - aiming only to do what is possible."

Here is where I disagree. I don't think 1) is true. Mathematics doesn't purport to convey absolute meaning. Mathematics knows its limitations and works within those limitations. Maybe naive mathematicians think mathematics reflects some absolute truth about our reality but I have a feeling that idea has gone by the wayside among category/set theorists and philosophers of mathematics (I think generally the idea that humans can know truth has gone by the wayside..)

I agree that mathematicians agree that absolute truth is not possible. And that, to a degree, this is reflected in their behaviour.

Neither mathematics nor mathematicians can do what is impossible. Whatever it is that mathematicians do must be what is possible. In this regard, mathematics and reality have no choice but to be aligned to a large degree. However, I feel that this alignment is despite mathematicians rather than because of mathematicians.

This is specifically reflected in the structure of axiomatic mathematics. I propose that there are two elements to axiomatic mathematics:

i) The Turing Machine equivalent in which a set of symbols can be manipulated in a deterministic fashion.

ii) The perceived meaning of that manipulation for a given set of symbols.

Part i) is unambiguous, definite and completely meaningless by itself.

Part ii) is where axiomatic systems gain their significance and usefulness (meaning)... but lies entirely outside of mathematics, being utterly undefined by any formal mathematical system.

Any Universal Turing Machine can emulate/simulate any axiomatic system. ZFC is just a particular Universal Turing Machine (equivalent) and has no more or less significance than any other UTM. A UTM is just a sufficiently flexible language. It has no inherent meaning, no inherent significance.

When we attribute meaning to symbols we do so from outside mathematics. "set", "empty set", "contains", "cardinality", "choice" are not mathematically defined terms. Attaching these external meanings to mathematical symbols is arbitrary.

This is where I see mathematicians being inconsistent. The definite, but meaningless, manipulation of symbols is confused with the arbitrary assignment of "meaning" to give the impression that there is a definite manipulation of meaning.

The urge to hang on to an impossible ideal has lead mathematicians to seize upon "definite and unambiguous" manipulation of symbols as being a worthy goal in itself. This illusion is supported/justified by pretending that those symbols have inherent meaning - or that importing meaning from elsewhere can be done in a meaningful way.

3) I don't think math is doing that thing that you claim it shouldn't be doing i.e. point 1).

Obviously mathematics is not doing something that is impossible to do.

But... axiomatic mathematics by itself has no meaning. Literally.

So, meaning is imported into axiomatic mathematics. That meaning is then manipulated (via attached symbols) which supposedly results in something of some significance even though we don't, and can't, know what the original, imported, meaning actually was (beyond the happy accident of some shared experience).

Axiomatic mathematics requires that the symbols have an associated meaning in order to have any significance. This is the fundamental error of axiomatic mathematics. It is hanging on to the ideal that it is possible to specify the meaning for something.

It isn't that we can't know exactly what a set is... We can't know anything about a set at all (beyond its relationships to other objects that we (also) know nothing about).

Hmm... I'm sat here typing on a keyboard, looking at a monitor, with a glass of water by my side and telling you that it is impossible to know what any of those things are. You, no doubt, are reading this on a screen with a beverage in easy reach and wondering why I'm attempting to communicate if I think it is such a futile effort... It isn't that we can't communicate - it isn't that mathematics can't communicate... it is that mathematics is not understanding itself. An essential component of communication is to... communicate. Mathematics off-loads the definition of meaning and thereby abdicates responsibility for understanding how and why we perceive meaning in language.

I want to address some other things you're saying.

"the hierarchy of infinity is supposedly defined with respect to ZFC... despite such definition being impossible."

Why do you have an issue with this? You say defined with respect to ZFC. Is this not exactly what you're talking about with "relativistic mathematics?" I feel like everything in mathematics is defined relative to ZFC (or some other axiomatic system). Thus you can take those axioms and undefined things (sets) as the sort of floating point to which everything is compared. what's the issue here?

One issue is how floaty the floating point is... If the initial floating point is completely undefined - then any extrapolation from that point is also completely undefined. No progress has been made.

We could argue that the axioms of ZFC do have subjective meaning for humans and that this is sufficient to provide some degree of meaning to any extrapolation of the axioms. This argument has some merit... except that we can't state what that subjective meaning is. We can't measure or otherwise quantify how subjective that meaning is with existing tools. As much as we do perceive meaning with respect to words... we can't communicate that meaning (except by reference to common experience).

As such - no - this is not what I'm talking about with regard to relativistic mathematics. This is the break from 'conventional' thinking that is hard to grasp because we have lived for so long with the assumption that we can define a thing to some degree.

Mathematics divides the world into "things" and "relationships" ("states" and "functions"). It then attempts to describe various states and the mappings between them. This is (mostly) futile.

Language gives us no mechanism to describe something directly. Which means it gives us no mechanism to describe something. The only thing we are able to describe is the pattern of relationships surrounding a "thing".

This is how language works - and how thought works. We communicate patterns of relationships. We attach meaning to particular shapes of relationships. The nodes between relationships are completely beyond any possibility of us knowing directly.

Okay - so "a pattern of relationships" is a "thing"... we can communicate a pattern of relationships - we can communicate a "thing". And if a pattern of relationships gives us any degree of a foundation... then that is more than nothing and axiomatic mathematics does have something it can build on, albeit something that can never be completely fixed or absolutely understood. Hence axiomatic mathematics does work a bit.

Bear in mind that a relationship also cannot be defined except via a pattern of relationships. Thus, our basis for relativistic mathematics is "patterns of relationships". We simply do not have access to anything else with which to build. Language can only communicate relationships - and the constraints on language apply to any system of interactions of any type.

Also you say that you don't think mathematics should lean on other fields to provide a framework. I think this is a little bit of mathematical hubris speaking. Perhaps a hubris similar to that which leads mathematicians to think mathematics finds absolute truths (not meaning to offend you here..) I think you may have to realize that mathematics is on just as shaky ground as the English language is. As languages they have the same rules and limitations. The difference is that mathematics plays within those rules in such a way that it can very efficiently be used to discover certain types of things that people find interesting while english plays within those rules in such a way that it allows people to communicate efficiently in every day activity and certain kinds of conversations. It's the same way binary and computer code allows computers to communicate information in the shortest amount of time possible. different languages for different purposes.

(You are playing the argument - not the man (and being constructive in the process) - no offense seen)

I see mathematics as being a formalisation of language - a particularly rigorous and self-knowing approach to language. As such I agree that mathematics has the same scope and limitations as other languages (mathematics is probably a class of languages rather than a single language...). My problem with mathematics leaning on external systems is that there is a significant likelihood of hiding knotty problems in those external systems. Which is exactly what I think is happening. Inheriting meaning from systems external to mathematics means that meaning within mathematics isn't well understood. As good as mathematicians are at playing with symbols (very, very good, indeed), all that manipulation is only as significant as the meaning that is attached. And the meaning is left to other, much less rigorous and precise processes to determine.

More to the point - I don't think that there is any need to leave this essential component to other disciplines. I think that a full understanding of mathematics (language) is entirely within the remit of mathematics.

Descartes and solipsism put a damper on the idea of absolute knowledge being achievable... but obviously there is a third option between "absolute knowledge" and "nothing". By relying on other fields, I feel mathematics is abdicating its responsibility in this regard. If mathematics is a self-reflective language, then it should explore and understand precisely how communication of knowledge works and specify the capabilities and limitations of communication in as much detail as is possible. In this regard, at least, mathematics should be self-contained.