Is everything a metaphor?

[John K]

I'm not sure what is meant by 'metaphor' in this context.

A metaphor is a function of language and language is ultimately subjective. What it comes down to is how we distinguish between representation and object. We distinguish between things subjectively and so mathematics is technically a language. It describes the world only in terms of its relationship to us.

Mathematics can lead to new discoveries, but only in the event that the logic leads to a conclusion that fits with the prior model.

Take that Ti-doms.

Sounds like you probably side with those who hold that maths is a tautology, which would definitely be an odd sort of metaphor.

 

[Ren]

language is ultimately subjective.

Wat

Sounds like you probably side with those who hold that maths is a tautology, which would definitely be an odd sort of metaphor.

In what way do you mean, John?

 

[John K]

In what way do you mean, John?

Well one view of mathematical logic is that it simply restates in various ways the axioms upon which it is layered and adds nothing new to them - all the information expressed in a chain of mathematical logic is entirely contained completely in the axiomatic statements from that point of view. The logical manipulations simply reformulate it for a variety of different purposes.

It doesn't sound like a perspective that can be explored effectively from within maths itself, but needs a philosophical reach. My own gut instinct is that it isn't true, but I couldn't put that instinct into a well argued case.

 

[Ren]

Well one view of mathematical logic is that it simply restates in various ways the axioms upon which it is layered and adds nothing new to them - all the information expressed in a chain of mathematical logic is entirely contained completely in the axiomatic statements from that point of view. The logical manipulations simply reformulate it for a variety of different purposes.

It doesn't sound like a perspective that can be explored effectively from within maths itself, but needs a philosophical reach. My own gut instinct is that it isn't true, but I couldn't put that instinct into a well argued case.

That's very clear. Thanks!

 

[Ren]

@John K

Would Anomalous's claim that maths is subjective indicate sympathies with an intuitionistic conception of math?

What do you make of intuitionism?

 

[Anomalous]

Sounds like you probably side with those who hold that maths is a tautology, which would definitely be an odd sort of metaphor.

I'm not sure I have an official stance. I'm just throwing it out there. But yes, that would be a more succinct way of putting it.

Wat

Wat

 

[Ren]

Wat

I was trying to make sense of the claim that language was 'ultimately subjective'. I'm still not sure what that means.

If it means the ability for language is internal, then yes. But language itself isn't, otherwise we wouldn't be able to understand each other. In fact, if Chomsky is right, the very structure of language is universally shared.

 

[Anomalous]

I was trying to make sense of the claim that language was 'ultimately subjective'. I'm still not sure what that means.

If it means the ability for language is internal, then yes. But language itself isn't, otherwise we wouldn't be able to understand each other. In fact, if Chomsky is right, the very structure of language is universally shared.

My thinking is that language emerges from an interpretation of something and is therefore limited by that subjectivity. In theory if a language existed between two people, they'd have no way of clarifying that a particular word holds the same meaning for both of them, without reference to some sort of external object. It's meaning is based entirely on its relationship to other words.

I think there are common identifiable patterns that can be traced to universal traits. But then it could be argued that our framework for recognising such patterns is a kind of paradoelia.

 

[Ren]

My thinking is that language emerges from an interpretation of something and is therefore limited by that subjectivity. In theory if a language existed between two people, they'd have no way of clarifying that a particular word holds the same meaning for both of them, without reference to some sort of external object. It's meaning is based entirely on its relationship to other words.

I think there are common identifiable patterns that can be traced to universal traits. But then it could be argued that our framework for recognising such patterns is a kind of paradoelia.

Wow, yeah I see what you mean now. It's a pretty deep point, and I believe still debated by philosophers of language.

I'm attracted to the idea that the meaning of a sentence derives from its relationship to other sentences. (Not sure it can be reduced down to words.) Saussure called language 'a play of differences,' which seems to echo the view.

With regard to the passage I bolded, I think this is called the thesis of the indeterminacy of translation. The meaning we ascribe to sentences changes even in our own lives, all the time, and we're only rarely conscious of it.

 

[dragulagu]

I was trying to make sense of the claim that language was 'ultimately subjective'. I'm still not sure what that means.

If it means the ability for language is internal, then yes. But language itself isn't, otherwise we wouldn't be able to understand each other. In fact, if Chomsky is right, the very structure of language is universally shared.

Au fil du temps moderne

Language does change, over a period of time, subjectively. It is not a solid structure.

 

[dragulagu]

all the information expressed in a chain of mathematical logic is entirely contained completely in the axiomatic statements from that point of view. The logical manipulations simply reformulate it for a variety of different purposes.

However does that leave room for new axiomatic formulations? Considering new realities or extremities becoming the norm (eg. the case of Dark Matter).

 

[John K]

@John K

Would Anomalous's claim that maths is subjective indicate sympathies with an intuitionistic conception of math?

What do you make of intuitionism?

Intuitionism as far as I understand it says that maths is a human construct that has no existence in the world other than as this. So bye-bye all those Platonic and Pythagorean idealisations. It’s not something I’ve thought very deeply about but it’s an intriguing issue. Maths would then be subjective in the same way as any other symbolic representation of meaning is within @Anomalous 's thinking.

Given my previous comments in this thread let’s make it clear that in this one I'm taking it that the external, objective world exists and our minds have got access to it through our senses and the inferences we build on them, but we experience it consciously only through a model that our minds create of it in approximate real-time.

It seems to me that there is nothing conclusive that can be said, but there are indications that support the idea maths is representative of something that exists outside human minds. The evidence is:

• It can be used to predict the behaviour of many things in the external world with amazing accuracy.
• It does this consistently for anyone who has grasped the rules and applied them to the right data in the right way.
• It throws up surprises that show that some maths applicable in one physical context turns out to be applicable in completely different ones too.
• etc

I think that it is parallelling something about the intrinsics of certain aspects of reality with very great precision. That's not to say necessarily that we are discovering maths but it certainly feels like we are inventing a simulation language that has the potential to mirror those aspects of reality with arbitrarily close precision. How close depends on things that lie outside the realm of maths - the discoveries of science and our ability to express tham well in mathematical terms. That spills over into questions about how real are the laws of physics, etc.

Now on the other hand, pure mathematics is simply a game of logic looking inwards on itself. I think it may be arguable that because it is a logically closed system with constructed axioms and rules then there is a sort of artificial objectivity about that sort of maths, regardless of whether applied maths is completely subjective or not. Because there always has to be an axiomatic starting point though, it could be argued that these are of necessity expressed in language that itself is subjective.

My gut feeling is that in maths we are dealing with something that is a human construct but which is eerily echoing the way reality behaves. But maybe that's true of all our mental relationshipo with the world?

From the brief glance I've had at intuitionist maths, there are some intriguing aspects that I haven't yet connected to the main premise of subjectivity. The most in-you-face is it's replacement of classical mathematical logic with an alternative set of rules. The idea that ~~A = A is fundamental to traditional maths, but this is denied in intuitionism - in the sense that it isn't necessarily true. That means you cannot prove something is true by falsifying it's opposite, but have to use constructive methods instead. I don't really know much about the history of this, but it probably comes from the melt-down in set theory in the late 19th century, together with the problems of dealing with Cantor's 'sets' of different sorts of infinity.

 

[John K]

However does that leave room for new axiomatic formulations? Considering new realities or extremities becoming the norm (eg. the case of Dark Matter).

Of course - definitely. But these are not derived from the mathematical logic applied to an existing set of axioms according to the idea maths is a tautology.

 

[Ren]

Au fil du temps moderne

Language does change, over a period of time, subjectively. It is not a solid structure.

Yes, but that’s a truism. Problem is this vague use of the word « language » — it would be better to distinguish ability, use, grammar, vocabularies, etc.

And I don’t think « subjective » is the right word either. I keep seeing the word used in this thread but I can’t make sense of what the link is between language and subjectivity. Subjectivity is an epistemic concept—not sure how it applies to the linguistic domain.

 

[dZpADTLrPmX4c]

Rutabaga.

 

[niar]

Of course I agree that not everything is a metaphor. That said, I disagree about maths and physics -- there is plenty of metaphor in science, especially revolutionary science. In fact it's absolutely fundamental to progress. New ways of looking at the world, whether scientific or otherwise, often emerge first as metaphorical. If they succeed and the theory becomes the dominant one (e.g. Newtonian mechanics, Relativity, etc.) the metaphors tend to become literalised, i.e. dead metaphors. Today we don't think anymore of sentences like 'the Earth revolves around the Sun' or 'supermassive black holes' as metaphors, but they once were. Metaphors have lifespans.

The OP's question is a lot deeper than it may seem at first.

I know the response was deeper, that was the -half- joke of my post. You've made good points, I agree with everything you said, except in my opinion the correct word here and in the OP is 'hypothesis' not 'metaphor'.

 

[Ren]

I know the response was deeper, that was the -half- joke of my post. You've made good points, I agree with everything you said, except in my opinion the correct word here and in the OP is 'hypothesis' not 'metaphor'.

I know you knew : )

I think I would be inclined to keep to the notion of metaphor as distinct from hypothesis. Nowadays, let's say a physicist makes a hypothesis about quantum entanglement: ok, very good. But the first time Schrodinger used the phrase 'quantum entanglement' ('Verschränkung' in German, apparently) in 1936, a lot of scientists must have scratched their heads trying to make sense of the idea. Similar things must have happened with the history of the concept of wave function, time as the fourth dimension of spacetime, gravitation, etc.

So I like Kuhn's idea of drawing a distinction between normal and revolutionary science, and saying that normal science deals in hypotheses that are essentially dead metaphors which were once invented as strange (live) metaphors by revolutionary science.

Admittedly this is not quite the sense of 'metaphor' which everyone takes for granted, but I find it quite fruitful.

 

[dragulagu]

Of course - definitely. But these are not derived from the mathematical logic applied to an existing set of axioms according to the idea maths is a tautology.

I'm not sure if that is 100% correct, the possible existence of Dark Matter (for example) is derived from a set of axioms leading to the current description of the Universe. There are no axioms written yet to describe the Dark Matter phenomenon precisely but there is the underlying requirement that this new set of axioms should co-exist, correlate, co-describe, extend the description of the Universe with our current axioms. Unless our understanding of the Universe as it is, is based on the wrong or incomplete set of axioms.

It's a bit of a philosophical take on the question that what we currently have described of the Universe, by mathematical standards derived from Human logic, is or would be sufficient to formulate the Universe
in its totality.

Yes, but that’s a truism. Problem is this vague use of the word « language » — it would be better to distinguish ability, use, grammar, vocabularies, etc.

And I don’t think « subjective » is the right word either. I keep seeing the word used in this thread but I can’t make sense of what the link is between language and subjectivity. Subjectivity is an epistemic concept—not sure how it applies to the linguistic domain.

For the development of structure of a language, I do agree. That should be solid (even though it is not, language does evolve through generations).
However, how would you describe language @Ren. (to me, language at its base definition is subjective).

@Ginny maybe something for you?

 

[John K]

I'm not sure if that is 100% correct, the possible existence of Dark Matter (for example) is derived from a set of axioms leading to the current description of the Universe. There are no axioms written yet to describe the Dark Matter phenomenon precisely but there is the underlying requirement that this new set of axioms should co-exist, correlate, co-describe, extend the description of the Universe with our current axioms. Unless our understanding of the Universe as it is, is based on the wrong or incomplete set of axioms.

It's a bit of a philosophical take on the question that what we currently have described of the Universe, by mathematical standards derived from Human logic, is or would be sufficient to formulate the Universe
in its totality.

Ah! There are several levels of response to what you say - I'll have a go and see where it leads:

• One is that an axiom in mathematics is something taken to be self-evidently true, or taken to be true by definition. The idea that maths is a tautology means that another independent axiom containing new information cannot be derived from existing ones. On that basis an axiom in maths is not something that can appear as a conclusion at the end of a chain of reasoning - that would be called a theorum.
• The idea that maths is a tautology claims that there can be no new information in a chain of mathematical reasoning, only a transformation of that information contained in the axioms. I didn't say I agreed with that - I have my doubts, but that's because mathematical logic is based on axioms expressed with the open-endedness of language, and the chains of well executed, accurate reasoning can generate speculations and insights in a mathematician that fall outside the scope of the axioms. It could well be, though, that a hypothetical computer logic machine that develops a chain of reasoning based on strict enclosure within a set of formalised axiomatic statements and the mechanics of logical processing, is caught within a tautology.
• In my statement I'm talking strictly about mathematics, and the existence or non-existence of dark matter is not a mathematical issue, but one of physics. There is a conjecture in physics that some matter that is only subject to the gravitational force is present in the universe and is responsible for the way that visible matter is observed to behave. There are alternative conjectures to explain this behaviour, including the possibility that the laws of gravitation need modifying when the looking at the scale of galaxes and larger. Some form of dark matter is seen as the most likely of the alternative explanations, but the way it is expected to behave currently doesn't fit all the observed facts. This issue won't be solved axiomatically but by observation, hypothesis and testing, and directed speculation in a typical science research loop. The maths used in this process will build on top of the observational evidence - the purely mathematical axioms that underlie such analysis are really to do with the validity and effectiveness of the mathematical procedures and have little to say directly about the physical existence or otherwise of dark matter.

 

[dragulagu]

Ah! There are several levels of response to what you say - I'll have a go and see where it leads:

• One is that an axiom in mathematics is something taken to be self-evidently true, or taken to be true by definition. The idea that maths is a tautology means that another independent axiom containing new information cannot be derived from existing ones. On that basis an axiom in maths is not something that can appear as a conclusion at the end of a chain of reasoning - that would be called a theorum.

• The idea that maths is a tautology claims that there can be no new information in a chain of mathematical reasoning, only a transformation of that information contained in the axioms. I didn't say I agreed with that - I have my doubts, but that's because mathematical logic is based on axioms expressed with the open-endedness of language, and the chains of well executed, accurate reasoning can generate speculations and insights in a mathematician that fall outside the scope of the axioms. It could well be, though, that a hypothetical computer logic machine that develops a chain of reasoning based on strict enclosure within a set of formalised axiomatic statements and the mechanics of logical processing, is caught within a tautology.
• In my statement I'm talking strictly about mathematics, and the existence or non-existence of dark matter is not a mathematical issue, but one of physics. There is a conjecture in physics that some matter that is only subject to the gravitational force is present in the universe and is responsible for the way that visible matter is observed to behave. There are alternative conjectures to explain this behaviour, including the possibility that the laws of gravitation need modifying when the looking at the scale of galaxes and larger. Some form of dark matter is seen as the most likely of the alternative explanations, but the way it is expected to behave currently doesn't fit all the observed facts. This issue won't be solved axiomatically but by observation, hypothesis and testing, and directed speculation in a typical science research loop. The maths used in this process will build on top of the observational evidence - the purely mathematical axioms that underlie such analysis are really to do with the validity and effectiveness of the mathematical procedures and have little to say directly about the physical existence or otherwise of dark matter.

Good, so here we go.
Very roughly first take @John K

One is that an axiom in mathematics is something taken to be self-evidently true, or taken to be true by definition. The idea that maths is a tautology means that another independent axiom containing new information cannot be derived from existing ones. On that basis an axiom in maths is not something that can appear as a conclusion at the end of a chain of reasoning - that would be called a theorum.

1. It is not self-evidently true, (true by definition, yes). I don't get the independent axiom part. A theorem is always based on a set of axioms.

2. ? The idea that maths is a tautology claims that there can be no new information in a chain of mathematical reasoning, only a transformation of that information contained in the axioms.

3. The (non)-existence of Dark Matter is absolutely one of a mathematical issue.