Exploring the Physical Reality of Math

[SW VandeCarr]

does math = reality?

That's like asking if thought (conceptualizations) equals reality. Some concepts are testable models. Others are not. If models are testable, they can be falsified. This falsification can be formal (logical) or empirical (experimental/observational).

These threads tend to be exercises in mystification.

 

[wuliheron]

These threads tend to be exercises in mystification.

I'd say they tend to be more of an exercise in institutionalized avoidance and denial.

 

[Hippasos]

We made up the language, but did we make up the patterns and relationships the language describes?

apeiron I'm with You. It really seems that mathematics is the closest language we (humans) have developed with nature but how close is it to "really" "understand".

Close enough?

 

[Maui]

Hey guys, quick question, I was wondering about this and thought maybe you guys would know.

Do you think math is built into nature, or is it something we made up to describe the world around us? I'm a senior ME major and I've seen math all the way up through PDE's and such, but I still can't figure out whether it's physically real or just falls out from unit analysis.

Thanks,

Kevin

'Real' by definition is always only that which is experienced in some way. In that sense, everything we propose about bringing reality into a framework is just models, not reality which is experienced and which is not the models(it's highly unlikely that it'd be ever possible to describe reality reliably within the framework of any model - this doesn't necessarily follow from Goedel's work).

 

[brydustin]

Hey guys, quick question, I was wondering about this and thought maybe you guys would know.

Do you think math is built into nature, or is it something we made up to describe the world around us? I'm a senior ME major and I've seen math all the way up through PDE's and such, but I still can't figure out whether it's physically real or just falls out from unit analysis.

Thanks,

Kevin

We live in a real world so its no wonder that the logic we deduce from analysis would be in nature. The fact that analysis "found" truth does not change the truth value of reality. Put another way, we are a product of nature, so its no wonder that our "natural" minds would deduce truths which are reflected in nature. So yes, mathematics is real and natural, but we didn't "invent" it, we merely reflect on it. Similarly, there is a perfection which exists which governs nature, mathematics is only a surface attempt at describing this perfection of the universe.

 

[cosmographer]

math is as perfectly cultural and historical as any human achievement. and it is as real as anything else. the paper you calculate on, the textbooks, the operations of thought it conditions. all of those things enact and intervene in the real at any point they are made relevant to someone or something. I would argue that it is only once you buy into the platonic myth of reality being something at distance from you, description, perception and so on, that the question can even be posed in OPs terms.

in my view "math" does not exist in some ideal space that maps over "nature" in some ideal space. simply because no pure, ahistorical, alocal realms like these exist. doing math is one mode of locally enacting the real and sure enough the practice of math has it's own load of conditions and possibilities. you try to describe some phenomenon, you try to mathematize it - those are all operations that happen in and to the world. once you're done you, your instruments, your thought, your body, and the conditions of possibility have moved on. realities have changed.

a real problem might show up if if we can agree that physical reality is in some way a bubbling process. there is novelty in the universe, we don't need to go further into any physical account. so we have novelty which originates realities, no matter if one thinks this at micro- or macrophysical levels. the question that may be asked then is this: what is the reality that determines the background conditions for the emergence of novelty. what is the unoriginated portion of nature classically construed? and once we can think that unoriginated portion as the reality that all of process, novelty, physics are grounded on we can ask the question: how? how is the continuous emergence of process determined?

that's when you might, like Whitehead, feel the need for a technical term "god" in your philosophy of physics. whatever keeps the processes in the universe in-check so to speak, have them run according so as to guarantee continuous process of some kind. to come back to the question of math then one could speculate: whatever unoriginated reality the excessive activity of the universe we can observe is originated from - that reality might determine certain conditions of continuous process. perhaps ruling in analogy to mathematical rules and operations? god (not the god of religion) might at some point have been a mathematician of sorts.

 

[rustynail]

does math = reality?

No, math = human translation of nature's language.

 

[cosmographer]

No, math = human translation of nature's language.

who is this nature? and does it speak by itself? math is a tool for transforming realities. those realities don't preexist their making, as if the mathematician would shuttle back and forth between nature and humans to bring the holy word. it is much more mundane than that. it is a technology of thought that enables transformative work in and of nature, very useful indeed!

a great book on the invention of modern math is Reviel Netz' historical study of the emergence of formalist styles of reasoning in ancient greece. this review does a good job at extracting the significance of that invention from a rather complex book:

http://www.bruno-latour.fr/articles/article/104-NETZ-SSofS.pdf

 

[apeiron]

who is this nature? and does it speak by itself? math is a tool for transforming realities. those realities don't preexist their making, as if the mathematician would shuttle back and forth between nature and humans to bring the holy word. it is much more mundane than that. it is a technology of thought that enables transformative work in and of nature, very useful indeed!

a great book on the invention of modern math is Reviel Netz' historical study of the emergence of formalist styles of reasoning in ancient greece. this review does a good job at extracting the significance of that invention from a rather complex book:

http://www.bruno-latour.fr/articles/article/104-NETZ-SSofS.pdf

Thanks, a very entertaining reference. To me it confirms that there is this key turn of the mind where we go from imagining reality as a process (developing towards ultimate limits) and as existent (the limits are now what have been achieved and so are "the real").

So when the Greek geometers drew diagrams in the sand with a stick, they made this leap from a reality seeking its perfection to a belief in the existence of the perfect forms themselves. The Greeks of course were not so willing to make the same leap when it came to ratios, incommensurability and infinity. Infinity was still a limit on the process of counting. But mathematics later fixed that with Cantor, etc.

And now we find ourselves torn between the two views. The developmental or process view is clearly "the real" as it is rooted in the "imperfect materiality" which is our world. There are only ever triangles as scrawled in the dust. But the ideal forms - the emergent limit states - also have a claim to reality because they "can't be imagined not to exist". What is more definite and concrete than an ultimate limit (a boundary concept like a triangle)?

This ontological confusion is then compounded by the usual epistemological one - mistaking the map for the terrain. The desire is to deal with the substance~form dichotomy (materiality and its boundary states) by assigning reality and unreality to an epistemological division. So the world (being out there) is real, the maths (being in our heads) is unreal. Yet clearly this does not work because the maths is still really out there in some sense - as the boundaries, the limits, the constraints. The maths is more than just a potential fiction, a social construction due to restricted cognitive technologies.

The way I sort out this nest of confusion is first to accept the epistemological division (I think the "modelling relations" crowd in theoretical biology - Rosen, Pattee, Salthe - do the best job here). So nature, reality, is a constructed view. Both our notions about its materials and its laws, its substance and its form, are "in our heads" and justified by a modelling relation (so it is a process with its own purpose, its own needs, not some dispassionate god's eye view).

So reality, as far as we can know it, is our invention. That applies to our mathematical ideas about it, but also our "physical impressions" too. It is all a map.

However on the whole, it is a very good map - as it has developed within the self-refining tradition of metaphysical abduction, scientific induction and logico-mathematical deduction (Peirce's pragmatic triad!).

And then the ontological bit of the story. We can see that limits only actually exist in the sense that they are the boundaries to what exists. They are how far a process of development can go in some direction before asymptotically tending to a limit. So in fact they are the ontically unreal. The boundary remains always infinitesimally just beyond where reality can reach (so as to be able to be seen to enclose it fully).

Yet boundaries also have a real causality. At least if you are a process thinker, a systems science, you believe that there is such a thing as downward causation and even final cause. So forms can act as constraints that actually shape materiality. Maths exists "out there" as something real in the sense that there are forms (of the kind maths can describe) which have a causal role in the realm of the real.

I guess we have to ask the question then whether the set of forms that humans can imagine is a superset or a subset of those that reality can express. Sometimes it seems our inventions are more fertile - we can elaborate to create more imagined things than can actually exist. Other times, that maths is in fact quite impoverished. It is a pretty crude map of the terrain. More subtle things are going on than we have captured so far.

 

[cosmographer]

The way I sort out this nest of confusion is first to accept the epistemological division (I think the "modelling relations" crowd in theoretical biology - Rosen, Pattee, Salthe - do the best job here). So nature, reality, is a constructed view. Both our notions about its materials and its laws, its substance and its form, are "in our heads" and justified by a modelling relation (so it is a process with its own purpose, its own needs, not some dispassionate god's eye view).

So reality, as far as we can know it, is our invention. That applies to our mathematical ideas about it, but also our "physical impressions" too. It is all a map.

However on the whole, it is a very good map - as it has developed within the self-refining tradition of metaphysical abduction, scientific induction and logico-mathematical deduction (Peirce's pragmatic triad!).

And then the ontological bit of the story. We can see that limits only actually exist in the sense that they are the boundaries to what exists. They are how far a process of development can go in some direction before asymptotically tending to a limit. So in fact they are the ontically unreal. The boundary remains always infinitesimally just beyond where reality can reach (so as to be able to be seen to enclose it fully).

Yet boundaries also have a real causality. At least if you are a process thinker, a systems science, you believe that there is such a thing as downward causation and even final cause. So forms can act as constraints that actually shape materiality. Maths exists "out there" as something real in the sense that there are forms (of the kind maths can describe) which have a causal role in the realm of the real.

Thanks for your thoughts. Mistaking the map for the territory surely is one mistake we can no longer afford to make. This is not to say that the map does not relate to territories, after all relevant relations had to be laboriously extracted from the very territory itself so to speak (which in math is a territory of the knowledge worker inheriting a mathematical tradition, the experimental setup is a human mind that has been equipped to do mathematical things together with theories, tools, colleagues and so on). A move that you seem to make here that I would like to be a bit cautious of is assuming "nature" as a constructed "view", something that is "our map" which "on the whole is rather good". By that move you seem to already take the map and it's general adequacy for given. And later you seem to jump from the idea of the map to the ontic qualities of a general territory. I'd argue here that when you look at the mechanisms by which maps actually are made and put in circulation, you end up with a different imaginary. The objects that transport the forms (the maps) are very concrete artifacts that have to travel into situations to make a difference. They are crucial to "forms" and "limits" gaining any kind of reality.

So the "ontology" drawn up in a technical math paper might not travel well to the practice of unclogging your toilet. It might not be a map adequate to the territory at all. In an extremely reductionist way one might perhaps want to insist that, yes, some miniscule aspect of the ontology drawn up in the paper captures well an aspect of what ontically happens when you are confronted with a clogged toilet. But at what price? Before you have tried to find a "form" or a "limit" the action has moved on and you might decide that you should rather call a plumber.

My main point here would be that pre-formatted reality is excessive and eventing. The reality of the forms in the math paper might be something that can occupy your metaphysical imagination, but that kind of mapping is also local to the very event of your imagining it. I would like us to keep adding ourselves and our specific situations and conditions (ontic, ontological and epistemological) back into the imagination of what kind of reality math can do. That move makes math totally historical and contextual. But luckily we have made it possible for "forms" and "limits" to travel - papers, words, computers, the postal system and so on.

So the more interesting case for me would be that of a mathematician, who, having a body and mind encultured to do mathematical things, would use the mathematical paper to do stuff in the world. Argue for funding money, argue with it against colleagues who hold other views, use a copy of it to make a provisional support for a table leg that had was too short, or use it as a model to calculate an aspect of climate change that eventually makes more effective action possible on an international scale! Or a nonmathematician who uses a copy of the same paper for totally divergent local activities, perhaps her gets a papercut or constructs a political theory from the formula. That's the kind of reality I would prefer to give to mathematics. Not to look for forms "out there" but to compel ourselves to look for the efficacy of math as "internal" to any specific practice .

So, sure the map might not be the territory, and sure it shares relevant features with the territory, but it also does work to transform the territory it is made relevant to.

 

[cosmographer]

To put it shorter: I don't only want "nature" as a thought-map of object-modelling relations, but as "doer" of local realities that are transformed by a being equipped with certain object-modelling technologies. Locally the model counts as much as the body as much as the situation in determining the next transformation.

 

[apeiron]

By that move you seem to already take the map and it's general adequacy for given.

Not really as I have already stressed "fit for purpose". So there is a criteria for making a judgement. In general, the map of western science is intended to give control over nature, and it is pretty easy to see the technological advance that results.

The objects that transport the forms (the maps) are very concrete artifacts that have to travel into situations to make a difference. They are crucial to "forms" and "limits" gaining any kind of reality.

The physical expression of the maps is of some interest to a philosophy of science student, but not crucial to the intellectual enterprise of map-making. Not in my opinion.

So the "ontology" drawn up in a technical math paper might not travel well to the practice of unclogging your toilet. It might not be a map adequate to the territory at all. In an extremely reductionist way one might perhaps want to insist that, yes, some miniscule aspect of the ontology drawn up in the paper captures well an aspect of what ontically happens when you are confronted with a clogged toilet. But at what price? Before you have tried to find a "form" or a "limit" the action has moved on and you might decide that you should rather call a plumber.

This seems rather spurious. Math would seem directly applicable to the form of the plumbing and the limits of its performance. But it would be the engineer who designs pipes and needs to model flows who would have need of the kind of technical map you are suggesting.

If a design of toilet was always clogging, would you call a plumber or an engineer?

My main point here would be that pre-formatted reality is excessive and eventing. The reality of the forms in the math paper might be something that can occupy your metaphysical imagination, but that kind of mapping is also local to the very event of your imagining it. I would like us to keep adding ourselves and our specific situations and conditions (ontic, ontological and epistemological) back into the imagination of what kind of reality math can do. That move makes math totally historical and contextual. But luckily we have made it possible for "forms" and "limits" to travel - papers, words, computers, the postal system and so on.

I think you are missing something vital if you focus only on the syntactical representation and leave out the semantics. So the modelling relations approach makes the point that models are in active interaction with the world. They don't really exist in the sense we are talking about when they are not doing anything (as in a book never read).

And while an individual making meaning of some mathematical idea at some moment is local and particular, there is still also the general activity of mathematically representing the world that lives in a multitude of minds over many centuries. That is just as real a level of action (just look at how the planet has been transformed in a couple of thousand years).

So the more interesting case for me would be that of a mathematician, who, having a body and mind encultured to do mathematical things, would use the mathematical paper to do stuff in the world. Argue for funding money, argue with it against colleagues who hold other views, use a copy of it to make a provisional support for a table leg that had was too short, or use it as a model to calculate an aspect of climate change that eventually makes more effective action possible on an international scale! Or a nonmathematician who uses a copy of the same paper for totally divergent local activities, perhaps her gets a papercut or constructs a political theory from the formula. That's the kind of reality I would prefer to give to mathematics. Not to look for forms "out there" but to compel ourselves to look for the efficacy of math as "internal" to any specific practice .

This is too reductionist for me. I am making an argument at the general level. The whole point is to generalise away the kind of localised quirks which you want to bring into play here.

So yes, again it is of interest to the anthropologist to record the variety in specific practices. But this ends up butterfly collecting unless you then extract general theories about "the practice".

So, sure the map might not be the territory, and sure it shares relevant features with the territory, but it also does work to transform the territory it is made relevant to.

I think it is more accurate to say the purpose of the map is to control the territory. I don't think it has the aim of transformation. And transformation is actually impossible in any fundamental sense. We can't change the laws of physics.

 

[apeiron]

To put it shorter: I don't only want "nature" as a thought-map of object-modelling relations, but as "doer" of local realities that are transformed by a being equipped with certain object-modelling technologies. Locally the model counts as much as the body as much as the situation in determining the next transformation.

Again, I too stress the active and purposeful nature of modelling. But I think you keep jumping to an unwanted stress on the particular. Metaphysics is about systematic generalisation - the shedding of the details that obscure. It is a discourse that privileges the universal. (Or am I just old-fashioned )

 

[cosmographer]

Thanks for your concise replies. I see that quite a bit about my position needs to be fleshed out better. I'll have to come back to that in a moment when I have more time. Btw is it possible to pack quote and reply into one quote? So that I could reply to the full pair of my previous post plus your reply? I don't think I'm finding the right buttons here

 

[binbots]

I think math does not exist in nature. But it is the only way our brains know how to understand it. 1 + 1 = 2 true. But where in nature does 1 appear. You can say one apple, but that is millions of cells, billions of atoms. etc. There are no perfect cirlces or shapes in nature. We have simplified our environment to make it eaiser to understand.

 

[apeiron]

I think math does not exist in nature. But it is the only way our brains know how to understand it. 1 + 1 = 2 true. But where in nature does 1 appear. You can say one apple, but that is millions of cells, billions of atoms. etc. There are no perfect cirlces or shapes in nature. We have simplified our environment to make it eaiser to understand.

And equally, nowhere in reality do we observe pure formless stuff. The material or substance always comes formed as an object, an event, some particular arrangement.

So the same argument applies to both the form and the substance of reality, yet most people would seem to feel that it applies more tellingly to the notion of form (because forms are taken to emerge from substance in standard reductionist view).

 

[disregardthat]

And equally, nowhere in reality do we observe pure formless stuff. The material or substance always comes formed as an object, an event, some particular arrangement.

It is the formless "stuff" we observe, form is just the appearance of substance, in the way the mind interprets sensory experience.

-------------

Mathematics does not exist in any particularly meaningful way. It exists as an activity, as an idea (or set of ideas), or as the results of firing neurons, but these are very trivial ways of saying that mathematics exists. In any platonic manner it does not. Even the notion of mathematical truth is exceptionally different from truth as an epistemological or even logical term. In mathematics, we can easily switch the operational label "truth" to "a rule". Theorems are equally valid rules as they are true statements, for mathematical existence is no less of a rule than, say, mathematical operations.

The statement "1 + 1 = 2" or more illustrative "432 + 257 = 689" is essentially the result of a calculation, and we somehow call it "truth". Not that this is wrong, but we can't consider it more than a label. What would be more similar to "truth" are geometrical theorems, like "the triangle formed by the diameter of a circle to a point on its periphery is a right triangle". This seems like a true statement, like "water has the molecular formula H2O", but it is not more than the result of a different (from arithmetical) sort of calculation. We do not talk about "lines", "triangles" and "circles" in geometrical theorems any more than we talk about "1", "+" and "=" in arithmetical statements. Is "1 + 1 = 2" a statement about "+"?

The moment we stop thinking of a mathematical statement as a statement about something (as opposed to a calculation or rule), we will be less inclined to insist on an independent mathematical reality. Why would there be something to talk about when asserting a mathematical statement? Does it pop into existence the moment we assert our axioms? Or are they discovered once we imagine what we are calculating? I find the idea of platonic existence in mathematics strikingly vague and misguided.

 

[apeiron]

It is the formless "stuff" we observe, form is just the appearance of substance, in the way the mind interprets sensory experience.

That's the point. It is so easy, but so mistaken, to believe that the substance is out there in the world and the forms exist only really in our minds as useful abstractions.

Yes, the reality of substance is what materialism in a general way presumes. But when you consider things more carefully, you discover that our impressions of materiality are also a projection of our ideas about what is "out there".

Tables and chairs seem to be made of solid stuff. Yet we know they are just largely empty space, an arrangement of atoms (if we are viewing reality through the prism of particle physics). Or that they are merely excitations of fields (if we step back even further to view through the prism of quantum field theory). There is nothing really material there (except perhaps from some form - some organised network of interactions).

The problem, as I have said, is really to do with causality. We are willing to grant reality to material and efficient cause, but have decided that formal and final cause are just fictions of our mind.

So the substantial aspects of reality are fundamental and "out there" regardless of whether we humans are around to witness them, or model them. But the formal aspects are at best emergent, epiphenomal, and not physically a part of the causality. They become merely convenient fictions that we invent after the fact.

All this is even more obscured because while maths is a way of representing form, it in fact does this via atomistic construction (as do the various off-shoots of this method of simplification such as classical logic, Turing computation, information theory and statistical mechanics).

So maths founds itself on axioms - global constraints, general statements on what is being taken as true about a class of events, general definitions of the objects that exist and the operations that are allowed on them. A formal syntax is created. And then an endless variety of particular forms can be constructed from all the arrangements the syntax allows.

Again, this mirrors the ontic view of the world "out there" as being just a construction of material atoms, with the global forms being merely emergent and lacking any "higher" causality, such as a purpose or a downwards-acting constraint.

Philosophically, this is a messed up way of thinking (though it is a very effective way of modelling as Western science has proven).

For instance, this is why people take seriously questions like is reality really some kind of giant cellular automata, or a Matrix simulation, or a Tegmark ensemble, or a Boltzmann brain? If even the forms of reality seem constructable (that there are no real downward acting constraints), then reality "might just in fact be constructed".

So that is why it is quite important (in a philosophical context) to disentangle the presumptions that have been made along the line to create maths, science and technology.

There have been simplifications for the sake of efficient modelling. But philosophically, we can see that there is such a thing as too simple here.

 

[disregardthat]

Yes, the reality of substance is what materialism in a general way presumes. But when you consider things more carefully, you discover that our impressions of materiality are also a projection of our ideas about what is "out there".

I agree with that "material" is also a projection of our ideas of what's "out there". That's actually why I said "stuff", since it doesn't make sense to talk about what's "out there" ("ding an sich" and all that...). But accepting that there is such a thing as "out there", and not that "all that exists is in our mind" is a different thing. And for this we use "substance" as a meaningless term of "the world out of reach".

Of course, I would say that atoms are not in a fundamental way "out there" as opposed to "illusions" like the things and objects we perceive. Any description by language is equally statements about mental projections as they are statements about reality, depending on the context, and with the same meaning.

 

[apeiron]

The statement "1 + 1 = 2" or more illustrative "432 + 257 = 689" is essentially the result of a calculation, and we somehow call it "truth".

And what guarantees the truth of such statements? You have some generalised truth - the axioms - which underwrites the syntactic constuction of these localised truths, some particular calculation.

So it all starts with a human framing an axiom - claiming some general truth. And inventing the syntax to add back in particular constraints to produce a guaranteed true localised outcome.

Geometry seems different only in that "the real world" still supplies some of the constraints. There is a dimensionality that gets abstracted away in algebraic representations of the same ideas. But even geometry has in fact long managed to generalise away our "material sensory impressions of the world". Gone are particular constraints on dimensionality such as their number, their flatness, their particular scale (fractal geometry), their separation (topology generalises away distance). Time or change also gets generalised away early in the peice.

So the geometric notion of a triangle still imports a lot of real world constraints (such as that angles must add up to 180 degrees). Geometry has since learned to be even more general than the world "out there" so it can now describe "any world". Or just about. There is a syntax that can add back constraints (such as open or closed curvature) and so construct models of possible worlds.

The moment we stop thinking of a mathematical statement as a statement about something (as opposed to a calculation or rule), we will be less inclined to insist on an independent mathematical reality. Why would there be something to talk about when asserting a mathematical statement? Does it pop into existence the moment we assert our axioms? Or are they discovered once we imagine what we are calculating? I find the idea of platonic existence in mathematics strikingly vague and misguided.

So you are right if what you are saying is that the mathematics we do in our heads - the syntactical operations - are nothing like what reality does out there. There is nothing like these kinds of mathematical statements happening when electrons scatter or a whorl of turbulence forms. That is not how reality operates. And so maths is not physically real in that narrow sense.

But reality does "look mathematical" in that the patterns we can construct from syntactical operations on axiomatic truths can have a good correspondence to the patterns that self-organise out in reality (via a more holistic causality - a causality we do not fully represent).

And while the axioms of maths are too generalised to be real (our universe has a particular topology and so is more constrained than our geometrical generalities recognise), we can as I say add back constraints on topology so that the syntactical representation do become realistically self-organising. This is what we do in simulating turbulence or chaos for instance I would argue. We set up the right constraints and the right patterns emerge. Suddenly the maths and the reality seem in close correspondence again. The maths looks physically real.

So it is a horribly complex situation. Perhaps it can be said that maths starts off naively real - the constraints that are part of reality are unwittingly imported when we do things like framing the axioms of euclidean geometry (importing flatness, contiguity, etc). Then maths becomes increasingly unreal as it is realized the constraints are too particular and can be abstracted away. We head towards axioms that are maximally unconstrained - going together with the invention of the syntax to add constraints back in as a matter of additive construction.

So there is a system that is quite unreal (or which just has a vestige of reality in its axioms). But which can then be brought back towards reality by adding back in the constraints that again make it behave quite like the way reality behaves. The maths shows computationally emergent self-organisation as we see with, for example, neural networks, cellular automata, chaos simulations.

[The difference is nature, as a system, finds its own contraints (and has no choice what they are most probably) while humans with their mathematical models have to choose the constraints - it is not something the mathematical system can do for itself. Unless you perhaps set up a "realistic" evolutionary process as with genetic algorithms, etc.]

 

[disregardthat]

I'm not sure we are on the same wavelength here (and I'm not particularly comfortable talking about constraints), but I have been talking about mathematical reality as in platonic reality, not as in physical reality. Mathematical statements are never statements about the physical reality. Logical statements are neither statements about physical reality, but in mathematics we have in a much more radical sense not really "statements" at all compared to logical statements and statements of science. Statements, theorems as well as axioms, can be (perhaps more aptly) be considered as rules, and are not statements about anything.

(The manner of which we guarantee the "truth" of statements such as "2+3=5" does not operationally change in the way I consider them)

Take set theory. We talk about sets, of course. But we never define sets. In fact, we don't need a definition of sets. It is not only useless, but irrelevant for the mathematics which spawn from it (and it would even be futile to try to do so in set theory). We have ourselves simply a collection of rules to utilize. It is actually a very odd thing to say that set theory is "a theory of sets". "Sets" are not really something that is being talked about (mathematically that is, we can casually talk about sets outside the formalities of mathematics). Sets did not get their existence the moment we created set theory, and sets did not exist before we invented set theory (platonically). Mathematically they exist, but mathematical existence, as I mentioned before, might as well be regarded as another rule.

Geometry is not different in this manner. We don't need definitions of lines or points, they are taken as primitive notions subject to the employment of mathematical rules given by the axioms. It is important to distinguish between the mathematical part and the practical part in which the mathematical results are used for, say, modeling nature (or making a winning strategy in a board game).

I'm not entirely sure how to understand what you say about reality having a topology, but it must be clear as day that the physical nature can not have mathematical properties in any fundamental fashion. Mathematics will in this sense only serve as a tool in a scientific model of nature (where it will make sense to talk about the topology or the geometry of space). My point is really that space does not have a topology which we attempt to describe mathematically.

(A straight line from A to B in the geometry of space time is taken as the route a photon will take from A to B. This notion of a line in physical could of course be otherwise, changing "the geometry of space")

 

[apeiron]

I'm not sure we are on the same wavelength here (and I'm not particularly comfortable talking about constraints)

The source of the discomfort is probably because "constraints" are implicitly active and causal. So it goes against the spirit of reductionism where things either are, or they aren't, there is no need to limit things so that they actually are just "are", and the other things are in fact "aren't".

But the framing of an axiom is an active constraint on possibility. It is the mathematician saying: many things might be true, but I am asserting now that this precise thing is true (and so everything else follows).

, but I have been talking about mathematical reality as in platonic reality, not as in physical reality. Mathematical statements are never statements about the physical reality
Logical statements are neither statements about physical reality, but in mathematics we have in a much more radical sense not really "statements" at all compared to logical statements and statements of science. Statements, theorems as well as axioms, can be (perhaps more aptly) be considered as rules, and are not statements about anything.

You are saying that mathematics is just the syntax. The semantics is unnecessary. Whereas science and logic need something substantial(!) to ground their formal statements.

I agree that mathematics is pretty much just concerned with the development of correct syntactical operations. It seems to be quite removed from the messy business of real things. But still, mathematics must make reality (or at least our measurable impressions of reality) both its point of departure and also its eventual place of return.

So to get the enterprise of rule-making, etc, going - the exploration of the rich syntactical possibilities inherent in any chosen set of axioms - the axioms have to be formed. And what it seems safe to presume is something humans agree from discussing their collective experience of reality. Axioms may be our sharp departure point from reality, but they arise out of that reality (or our ontic beliefs) by the same token.

Then having elaborated itself in the platonic realm of pure ideas thinking themselves , mathematics must return to reality as modelling. There is a reason why human society values maths and it is not because there is something useful in endless syntactical noodling. Rather, it is Wigner's unreasonable effectiveness that makes maths valued. Exactly what kind of syntax will be useful is unpredictable (and much may indeed be useless), but the payoff in terms of being able to model reality is obvious enough.

Take set theory. We talk about sets, of course. But we never define sets. In fact, we don't need a definition of sets. It is not only useless, but irrelevant for the mathematics which spawn from it (and it would even be futile to try to do so in set theory). We have ourselves simply a collection of rules to utilize. It is actually a very odd thing to say that set theory is "a theory of sets". "Sets" are not really something that is being talked about (mathematically that is, we can casually casually talk about sets outside the formalities of mathematics). Sets did not get their existence the moment we created set theory, and sets did not exist before we invented set theory (platonically). Mathematically they exist, but mathematical existence, as I mentioned before, might as well be regarded as another rule.

I'm a bit confused here as I would have thought the story was that mathematics attempted to find a foundations in set theory. Then when that didn't work, it had to go looking for something more general (less constrained) in category theory.

To "exist" in the platonic mathematical sense, set theory would seem to have to be "self-evident" in some incontrovertible fashion. Just as reality is self-evident and resists our attempts to controvert its existence.

Set theory couldn't prove itself. And I would have thought all the business with Russell and Godel was evidence that maths isn't actually platonic and needs reality as its at least vestigal departure point.

Also, in saying things like ideas exist "before" we think them, well they exist as concrete possibilities rather than actually existing. That would be the realist position. The platonic position would indeed say that the ideas exist outside of time itself. They are immutable (changeless) and so therefore "eternally present..or not present" of necessity. It is truth that creates a mathematical idea, and the untrue idea cannot exist.

This platonic statement sounds convincing. Until you come back to the fact that it all has to start somewhere. That in fact our axioms, our self-evident truths, are rooted in our very human impressions of reality. Mostly this fact can be avoided as people rarely discuss axioms in a philosophical way. They just assume them and get on with the game of syntactical elaboration.

I'm not entirely sure how to understand what you say about reality having a topology, but it must be clear as day that the physical nature can not have mathematical properties in any fundamental fashion. Mathematics will in this sense only serve as a tool in a scientific modeling nature (in which it will make sense to talk about the topology or the geometry of space). My point is really that space does not have a topology which we attempt to describe mathematically.

What I said was that geometry/topology has been generalised to the point where it no longer tries to describe our reality, but describes any kind of "world" as a somehow connected space or set or relations. Then to use the mathematics to describe/model our own world, we have to add back some of the constraints that have been relaxed.

So the real world has organisation. It has particular global constraints that exist! They may have developed, they may be still dynamic and slowly changing, but they are definite and persistent enough that they seem to define our universe. In maths, we have stripped away everything that seems particular so as to arrive at the most general. And so to model reality, we have to do the (unnatural, artificial) thing of adding constraints back to simulate the actual organisation of reality.

OK, says our mathematician God, I need to construct me a world. Give me just three spatial dimensions. Toss in a few more perhaps to make some stringy particles. Let's inflate this thing so big its got to look largely flat. I need a few constants and a big entropy gradient. And dang, I baked me a universe.

Reality itself would have arisen quite differently - not constructed by some unconstrained being but self-organised via the development of a particular set of constraints. And as I say, that self-organising story is tough to model because that is not the mental tool-kit we have been developing the past 2500 years.

You might actually need a maths that is a bit different in spirit. One that can model the development of global constraints rather than one where the mathematician stands outside and tosses constraints into the cooking pot to see what happens.

 

[disregardthat]

Set theory is above anything else a common context for mathematics, a "playground" in which most of mathematics can be formalized. We have various contexts for mathematics, and the foundational aspect of them are purely operational. Axioms are pre-existent as much as a building exists before it is built, a very trivial sense. The possibilities were there, but in mathematics we choose to go forth in any way we please by extending our calculus with additional rules.

Within set theory this aspect of mathematics is very much reduced, since we are constrained by the axioms. But then there are extensions: such as the existence of inaccessible cardinals, exploration of different set theories and conflicting set theoretical axioms, categories, and so forth. Ultimately we are after all not so much constrained by axioms, which can be created on the spot, but a common "playground" has its obvious benefits. Obviously set theory is a recent invention, and how mathematics consisted of various extensions is more easily seen before set theory. Such as the invention of Calculus, which drastically expanded the mathematical discourse of the time.

No axiom of mathematics is "self-evident" in any meaningful fashion. How can it be? Mathematics is not evident at all, the practical aspects must not be confused with the purely formal nature of mathematics (and then not necessarily by symbols, we can easily have a mathematical calculus using words, but there is no denying that what governs the calculus in question are formal rules). While I'm no historian; historically all of arithmetic (and obviously geometry) were stated in words, and symbols were a "tool" for expressing what needed to be said in words, which was considered to be the rigorous method. If this is what you mean by semantics, it is clearly incorporated in the formal aspect of mathematics.

Mathematics in scientific contexts conserves their purely formal nature, but the transition from mathematical to scientific statements is subtle but important. It is vital to understand that this transition is not mathematical, but purely logical. In Newtonian mechanics when the math is done, the result is treated as a physical statement. The transition consists of this interpretation. It is logical because we are then treating it as a logical statement within our model, and then ultimately as a scientific statement if it is translated to a statement of physical nature subject to observation and validation etc.. F = am can be considered a physical statement or a mathematical one, depending on context, one referring to a relation between physical properties, and another one to the mathematical aspect of calculation. I believe I have a more restricted view of what mathematical activity is than you! But I agree with you that this transition is the important part, which in many cases serves as a concatenation of the logic in physical models.

 

[apeiron]

Ultimately we are after all not so much constrained by axioms, which can be created on the spot, but a common "playground" has its obvious benefits.

I was meaning that your axioms are the constraints which create the playground. They limit vague possibility in a fruitful way so that there is a definite kind of play going on.

You would agree that axioms define what is legitimate in the playground? So that is what constraint means here.

Obviously set theory is a recent invention, and how mathematics consisted of various extensions is more easily seen before set theory. Such as the invention of Calculus, which drastically expanded the mathematical discourse of the time.

I would have said calculus is in fact a particularly murky example of how mathematics develops . People were groping around for a long time for the foundational justification of the early guesses that appeared to work.

But what you see as "extensions", I see as generalisations or relaxations. So the relaxing of some constraining assumption - like the the impossibility of infinitesimal quantities or the taking of limits - is what opens up the new terrain.

No axiom of mathematics is "self-evident" in any meaningful fashion. How can it be?

OK, this is arguable because it is said it is no longer required of mathematical axioms. But clearly it was originally a requirement. And even today there has to be some motivation to propose an axiom and some conviction that it is not self-contradicting.

From wiki...
http://en.wikipedia.org/wiki/Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...

but...

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

If this is what you mean by semantics, it is clearly incorporated in the formal aspect of mathematics.

No, semantics is what gives meaning to the symbols - whether they be notation or words.

Mathematics in scientific contexts conserves their purely formal nature, but the transition from mathematical to scientific statements is subtle but important. It is vital to understand that this transition is not mathematical, but purely logical. In Newtonian mechanics when the math is done, the result is treated as a physical statement. The transition consists of this interpretation.

Yes, someone interprets the formalism in terms of physical values. There is the semantics that gives meaning to the syntactical operations. The measured inputs and outputs that animate the equation.

And in modelling relations theory, this act of interpretation is held not be logical but in fact the informal part of the business. The meaning of the mathematical statement would seem "logical" - or rather obvious to the person with the right training. But it is not actually logical in the sense of being also a formalisable operation. Reducible to syntax and so absent of semantics.

I believe I have a more restricted view of what mathematical activity is than you! But I agree with you that this transition is the important part, which in many cases serves as a concatenation of the logic in physical models.

I don't think there are any wild disagreements here. The way maths relates to reality is a very intricate business, obscured by its own social history. That is why the OP, seemingly so simple, sparks such long replies.

 

[apeiron]

Wiki suggests "primitive notion" more accurately captures my meaning here than "axiom". At least the informal/semantic basis of maths is more openly admitted in this usage.

In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or other formal system, the role of a primitive notion is analogous to that of axiom.

http://en.wikipedia.org/wiki/Primitive_notion

Examples given:

In Naive set theory, the empty set is a primitive notion. (To assert that it exists would be an implicit axiom.)

In Peano arithmetic, the successor function and the number zero are primitive notions.

 

[octelcogopod]

This ontological confusion is then compounded by the usual epistemological one - mistaking the map for the terrain. The desire is to deal with the substance~form dichotomy (materiality and its boundary states) by assigning reality and unreality to an epistemological division. So the world (being out there) is real, the maths (being in our heads) is unreal. Yet clearly this does not work because the maths is still really out there in some sense - as the boundaries, the limits, the constraints. The maths is more than just a potential fiction, a social construction due to restricted cognitive technologies.

The way I sort out this nest of confusion is first to accept the epistemological division (I think the "modelling relations" crowd in theoretical biology - Rosen, Pattee, Salthe - do the best job here). So nature, reality, is a constructed view. Both our notions about its materials and its laws, its substance and its form, are "in our heads" and justified by a modelling relation (so it is a process with its own purpose, its own needs, not some dispassionate god's eye view).

So reality, as far as we can know it, is our invention. That applies to our mathematical ideas about it, but also our "physical impressions" too. It is all a map.

However on the whole, it is a very good map - as it has developed within the self-refining tradition of metaphysical abduction, scientific induction and logico-mathematical deduction (Peirce's pragmatic triad!).

And then the ontological bit of the story. We can see that limits only actually exist in the sense that they are the boundaries to what exists. They are how far a process of development can go in some direction before asymptotically tending to a limit. So in fact they are the ontically unreal. The boundary remains always infinitesimally just beyond where reality can reach (so as to be able to be seen to enclose it fully).

Yet boundaries also have a real causality. At least if you are a process thinker, a systems science, you believe that there is such a thing as downward causation and even final cause. So forms can act as constraints that actually shape materiality. Maths exists "out there" as something real in the sense that there are forms (of the kind maths can describe) which have a causal role in the realm of the real.

I guess we have to ask the question then whether the set of forms that humans can imagine is a superset or a subset of those that reality can express. Sometimes it seems our inventions are more fertile - we can elaborate to create more imagined things than can actually exist. Other times, that maths is in fact quite impoverished. It is a pretty crude map of the terrain. More subtle things are going on than we have captured so far.

I really enjoyed this post. The first thing that comes to mind (especially regarding the superset/subset question) is that it would seem logical that by abstracting reality into shapes these abstractions in the mind would in fact be a superset. If they would be a subset it implies to me that there is some aspect which will never be reachable and imaginable since they would either have to be equal to, or "above" reality to encompass reality, right?
It could be the barrier is simply the fact that the abstracted models are not actually physical(in the normal physical definition), but in that sense the models and map could be identical to reality? In my mind it becomes a bit confusing after awhile I must admit.

 

[apeiron]

I really enjoyed this post. The first thing that comes to mind (especially regarding the superset/subset question) is that it would seem logical that by abstracting reality into shapes these abstractions in the mind would in fact be a superset. If they would be a subset it implies to me that there is some aspect which will never be reachable and imaginable since they would either have to be equal to, or "above" reality to encompass reality, right?

Yes, we need to "rise above" - make generalisations, find universals, abstract, relax constraints - to model. Maps should contain the least information needed to do the job. And a triangle can be seen as a map of triangular relationships in general. There is something definite by which to measure reality's approach to a maximally reduced ideal form.

But then where does this ideal exist? The traditional debate says either the ideal is just a construction of the human mind (so is just a choice and not real in any useful sense of the word) or else it is transcendent - outside our reality and also inhabiting its own Platonic realm (with its own laws of operation and existence).

Whereas I see a story inbetween these extremes of convenient fiction and transcendental entity. A triangle would stand as a limit state on reality. It is more than an idea (because physically, assuming a euclidean spacetime, it is not as if we could have imagined something different being true as the limit description). But it is less than transcendent as it is only the limit of reality (of the universe as it actually is). It emerges as part of this physical existence and so exists (in the non-existent way that boundary exist!) only because there was a reality that could have this kind of limit.

We can then of course generalise our early ideas about triangles and Euclidean space that seem to be an accurate description of actual reality-limits. We can relax some of the constraints which were taken as axiomatic and discover curved geometries, for example. And then reality can turn out to be that way too.

We can keep on generalising, throwing away constraint after constraint and seeing what is left. Perhaps we are following the same path as reality takes, perhaps not. It is not necessary that this is the right path, on the other hand, it probably is. Depends on how good we are at identifying the constraints and choosing what to throw out.

 

[Ken G]

I think the subset/superset dichotomy is an interesting one to consider. Like most dichotomies, we should anticipate two aspects from the start:
1) The dichotomy identifies separate directions, more so than separate possibilities. We should not expect an either/or kind of answer.
2) The truth emerges as a kind of combination of, or an interaction between, the extremes, not only because the truth borrows a little from both, but also because there is a kind of essential tension there.
In the triangle analogy, this might play out as saying that on one hand, a perfect triangle is a subset of geometric forms, and the triangles we "find" in reality are less perfect, so the perfect example seems like a subset. But then we also recognize its perfection stems not from being a particular example, but rather from being a limiting example. So in that sense, they are a superset-- something outside of reality that reality never reaches. So there is a kind of "Platonic" element to a triangle in the way it forms a kind of boundary to the actual. Shall the closure of the actuality count as part of the reality? That's a similar question as to whether the imaginations of the mind count as a subset or superset of what is real.

On a related matter, I'm reminded of what I tell my daughter when she asks me if unicorns are real. She loves unicorns, so if I told her "no, you love a fiction", I would not be serving her. So instead I say "the way you feel about unicorns is certainly real, although there is in actuality no such animal as a unicorn." That she can imagine unicorns, and generate a strong feeling about them, is a part of the reality of her relationship with unicorns, so she has in a sense stretched reality to include something it might not otherwise have included. Perhaps it is the same with triangles, and mathematics is yet another of reality's many creation processes.

Of course, then people always ask, "does that mean Newton's laws didn't work before Newton?" My answer to that is "obviously yes-- before Newton there were no laws to test whether or not they worked. However, we can take data from before Newton, and, after Newton, test that Newton's laws work on that data." So Newton's laws only work after Newton, but they work on data from before Newton. After Newton, the reality expanded to include Newton's laws, and whatever extent those laws work, and that extent applies as well to data from before Newton as after.

 

[disregardthat]

I would have said calculus is in fact a particularly murky example of how mathematics develops . People were groping around for a long time for the foundational justification of the early guesses that appeared to work.

My opinion is on the contrary, the extensions made were excellent example of mathematical progress. Creating new mathematics is essentially what we do, but with set theory we have captured two things: 1) a formalization of the type of extensions we are inclined to do, and 2) an axiomatic setting in which mathematics can be used. Both of these are useful, but not essential for mathematics.

But what you see as "extensions", I see as generalisations or relaxations. So the relaxing of some constraining assumption - like the the impossibility of infinitesimal quantities or the taking of limits - is what opens up the new terrain.

Extension and generalizations may often the same thing, and some times they may not. A trivial example may be defining 0^0 = 1 in some particular setting. A little less trivial example (which may be thought of a generalization with some good will I guess) is giving sense to 0.3333... Originally when dealing with integers and then fractions, 0.3333... didn't have any meaning, and it wasn't deduced that 0.333... = 1/3. It was rather decided that when a sequence of digits reappear in the division process, we connect the sign "..." to the end of a recurring sequence of digits to represent the fraction in question. Later it was taken as an infinite sum. We extended our mathematical notation, which is equivalent to extending mathematics (yes!).

OK, this is arguable because it is said it is no longer required of mathematical axioms. But clearly it was originally a requirement. And even today there has to be some motivation to propose an axiom and some conviction that it is not self-contradicting.

It doesn't make sense to convinced of a axiom, it doesn't even make sense to be "convinced" of any particular part of mathematics. For mathematical statements, it is like building a toy model by the step-by-step guide and say about the final product "this is correct." More appropriately we will say "this was done correctly", and that is an entirely different thing. What does it mean to be convinced of "502+32 = 534"? We really aren't convinced of this statement (our terms treat it as such (a statement) which makes it sound like we are, but that I covered when I talked about mathematical truth being more of a label). Rather, we are convinced that we calculated correctly, which again is an entirely different thing.

In the same sense, being convinced of an axiom does not make sense. We didn't calculate anything, but we gave rule. For axioms, it is like being convinced of the rules of chess. We never say "the rules of chess are correct", but we may say "these are the rules of chess", which "translates" to "these axioms are correct". And we may say "this move is correct", corresponding to a theorem, meaning it was in accordance with the rules (the axioms).

I am aware of the historical context of this, but their conviction was not of the mathematical statements/axioms themselves, but rather of what they modeled. They were convinced that geometric calculations captured measured lengths, or that adding two integers would give the total number of stones split in two piles etc...

Both historically and currently, the view that euclidean geometry is "true" per se is nonsense. There are no physical lines or points to speak of, and there are no platonic lines or points to speak of. And by the same token general relativity is neither "true". (And that general relativity refutes euclidean geometry is meaningless in any interpretation i can think of)

And in modelling relations theory, this act of interpretation is held not be logical but in fact the informal part of the business. The meaning of the mathematical statement would seem "logical" - or rather obvious to the person with the right training. But it is not actually logical in the sense of being also a formalisable operation. Reducible to syntax and so absent of semantics.

I have a slightly broader understanding of the term "formal". It is the language we use into argue mathematically that is formal. In this sense semantics is also formal, and not opposed to, say, formal operations.

I think it is a trap to think of logic as "formalized rules of inference". Logic is part of the structure of language, which is critically important, and not rules in essence. Yes, logic may well be formalized, but how then are we arguing for more general logical statements following logical axioms? We are using the logical axioms much like we are using mathematical axioms, and that is just what is going on: we are doing mathematics. We are using logic to argue about logical rules. Logic, in a sense, "hovers" above all the formalizations regardless of whether it itself is what has been formalized. In this sense I am still convinced that the transition from mathematical calculations to a statement in a given physical model is logical in nature.

 

[qsa]

in the beginning there was no PI then the integers said let there be PI and every other matter(pun intended). so the universe was created with six constants h,c,e,alpha,Me,Mp. apeiron check your PM maybe you will see the light (also pun intended). and spread the gospel.

 

[apeiron]

Extension and generalizations may often the same thing, and some times they may not.

Does "extension" have some technical meaning here? I'm sure you don't intend the set theoretic usage.

To make it clear, I would be talking about the dichotomy of induction and deduction - which are definitely different, if complementary, operations. So you induce to generalise, deduce to recover particulars.

A trivial example may be defining 0^0 = 1 in some particular setting. A little less trivial example (which may be thought of a generalization with some good will I guess) is giving sense to 0.3333... Originally when dealing with integers and then fractions, 0.3333... didn't have any meaning, and it wasn't deduced that 0.333... = 1/3. It was rather decided that when a sequence of digits reappear in the division process, we put the sign "..." to represent the fraction in question. Later it was taken as an infinite sum. We extended our mathematical notation, which is equivalent to extending mathematics (yes!).

This is somewhat confusing as it sounds like you are saying decimal notation would have come before fractions in the history of math.

I would say first came the idea of "cutting into three equal parts". So the "inductive" part of the argument was the generalisation that any whole can be divided equally. And the deductive part was that dividing into 3 equal bits would be a particular example of this operation.

Then along came decimals and suddenly dividing 10 by 3 was an issue. One solved by ... to stand for an infinite process taken to have a limit. But is this an "extension", something done specially just to deal with this situation, or the adoption of a generalisation about taking limits that had arisen already. I mean, that it was the right thing to do was not deduced from fractional notation but from broader principles.

It doesn't make sense to convinced of a axiom, it doesn't even make sense to be "convinced" of any particular part of mathematics.

You are saying I think that mathematicians should be free to believe anything. But as the reference to primitive notions shows, in practice your axioms have to be quite convincing to your peers in some sense.

And remember that the OP is about the exact status of maths vs physical reality. And so we have to become very careful and precise about the actual practice of maths. I've provided references to that actual practice. So you would have to show me that some of the important and useful axioms of maths arose as a result of no-one having any prior conviction about their worth or ontic validity.

What does it mean to be convinced of "502+32 = 534"? We really aren't convinced of this statement (our terms treat it as such (a statement) which makes it sound like we are, but that I covered when I talked about mathematical truth being more of a label). Rather, we are convinced that we calculated correctly, which again is an entirely different thing.

But now you are not talking about the axioms but the calculations. So yes, an entirely different thing.

In the same sense, being convinced of an axiom does not make sense. We didn't calculate anything, but we gave rule. For axioms, it is like being convinced of the rules of chess. We never say "the rules of chess are correct", but we may say "these are the rules of chess", which "translates" to "these axioms are correct". And we may say "this move is correct", corresponding to a theorem, meaning it was in accordance with the rules (the axioms).

The rules of chess would indeed be a set of constraints. And for the chess pieces, they would appear to be the necessary truths of their world . But from our human point of view, we can see the rules are arbitrary. Or "axiomatic" only in the sense that the rules must be such that we can derive pleasure or diversion from their existence.

Our relationship to the world is a little different. We seem to have little choice about its rules. So - unless you are arguing that maths is just a diverting game, and does not have a primary sociological purpose of modelling reality - the chess analogy falls a little flat.

I have a slightly broader understanding of the term "formal". It is the language we use into argue mathematically that is formal. In this sense semantics is also formal, and not opposed to, say, formal operations.

Well, going back to epistemology, the map is not the territory. And the map is pure syntax. It cannot understand itself. The semantics lies elsewhere in the informal part of someone reading a map to find a path across the territory.

I agree, many believe that semantics can be reduced to syntax. Every computer scientist, for a start. But that is another (lengthy) discussion.

I think it is a trap to think of logic as "formalized rules of inference". Logic is part of the structure of language, which is critically important, and not rules in essence. Yes, logic may well be formalized, but how then are we arguing for more general logical statements following logical axioms? We are using the logical axioms much like we are using mathematical axioms, and that is just what is going on: we are doing mathematics. We are using logic to argue about logical rules. Logic, in a sense, "hovers" above all the formalizations regardless of whether it itself is what has been formalized. In this sense I am still convinced that the transition from mathematical calculations to a statement in a given physical model is logical in nature.

Yes, and now you are agreeing that symbols must be grounded, that syntax is not semantics.

Logic can't formally generate itself. We need to be convinced of some primitive notions to get the game going.

 

[Ken G]

And in this context we should not forget the Godel proof that semantics and syntax can never be exactly the same thing (no matter what the computer scientists think!). In any interestingly complex and consistent system of axioms, there will always be at least one semantic truth that is not a syntactic truth. So we have a wedge there, although how wide it is is not at all clear-- it might be the most imperceptible crack, or it might be a vast chasm that we simply have not the mental clarity to see.

 

[disregardthat]

Does "extension" have some technical meaning here? I'm sure you don't intend the set theoretic usage.

As I mentioned briefly later, extensions of a mathematical calculus is essentially creating new notation. What do you mean by set theoretic extensions?

This is somewhat confusing as it sounds like you are saying decimal notation would have come before fractions in the history of math.

It may have been somewhat chronologically confusing, but what I meant was that after we have established the digit representation of fractions, recurring sequences were given a meaning by adding "..." to the end of a recurring sequence to denote the corresponding fraction (from which the digits were calculated). The point is that we extended our calculus to incorporate symbols such as 0.333..., and did not finally discover or deduce what it actually was, as if it had any meaning prior to our decision.

You are saying I think that mathematicians should be free to believe anything. But as the reference to primitive notions shows, in practice your axioms have to be quite convincing to your peers in some sense.

Convincing in what sense? I don't believe any non-platonist out there "believes" in "sets" or their properties any more than one would believe in any abstract concept. He may were well believe in that mathematics ought to be extended in accordance to set theory however, but this is of course an entirely different thing.

And remember that the OP is about the exact status of maths vs physical reality. And so we have to become very careful and precise about the actual practice of maths. I've provided references to that actual practice. So you would have to show me that some of the important and useful axioms of maths arose as a result of no-one having any prior conviction about their worth or ontic validity.

The problem is what does "ontic validity" mean for mathematics. I propose it is utter nonsense, and while many may have convictions of something, it is certainly not ontologically in nature. The question that doesn't seem to be answered is "what is mathematics talking about"? Without a proper answer for this we can't start talking about the ontology of mathematics.

But now you are not talking about the axioms but the calculations. So yes, an entirely different thing.

Yes, I first tried to explain how we more appropriately could consider mathematical statements, and then axioms in the second paragraph.

The rules of chess would indeed be a set of constraints. And for the chess pieces, they would appear to be the necessary truths of their world . But from our human point of view, we can see the rules are arbitrary. Or "axiomatic" only in the sense that the rules must be such that we can derive pleasure or diversion from their existence.

Our relationship to the world is a little different. We seem to have little choice about its rules. So - unless you are arguing that maths is just a diverting game, and does not have a primary sociological purpose of modelling reality - the chess analogy falls a little flat.

The question here is not the purpose of mathematics, but its status as a whole, as you mentioned, is the question here. Thus being arbitrary or not is not that relevant. Axioms are created for various purposes: to use mathematics in various circumstances, but this is not the issue here. That we are more inclined to, say, make axioms for physical models doesn't mean that mathematics have any connection to physics. If all mathematics were used for was to analyze games in particular, would this establish some sort of connection with games (other than it is actually what it is used for)?

 

[apeiron]

The question here is not the purpose of mathematics, but its status as a whole, as you mentioned, is the question here. Thus being arbitrary or not is not that relevant. Axioms are created for various purposes: to use mathematics in various circumstances, but this is not the issue here. That we are more inclined to, say, make axioms for physical models doesn't mean that mathematics have any connection to physics. If all mathematics were used for was to analyze games in particular, would this establish some sort of connection with games (other than it is actually what it is used for)?

Whether maths is arbitrary, platonic, or something else inbetween is the whole point of the OP.

We can quickly agree there is an absolute epistemic divide - the map is not the territory. But that still leaves three burning questions.

1) What is the nature of the map? (How is it formed, how does it operate?)

2) What is the nature of the relationship of the map to the territory? (Modelling relations for example takes the specific view that measurement, and so the grounding in semantics, is informal).

3) Then what is actually "out there"? Is the territory in any sense "mathematical" too?

Your replies are muddling all these issues.

What you appear to want to argue is that maths is so intrinsically arbitrary that there is no necessary connection to reality - either as ontology or even epistemology. Math is not even modelling, just some kind of abstract noodling. Beliefs and their consequences played out without any purpose or grounding (though if scientists and philosophers somehow find a use for these arbitrary mind games, well, mathematicians will smile benevolently).

 

[apeiron]

And in this context we should not forget the Godel proof that semantics and syntax can never be exactly the same thing (no matter what the computer scientists think!). In any interestingly complex and consistent system of axioms, there will always be at least one semantic truth that is not a syntactic truth. So we have a wedge there, although how wide it is is not at all clear-- it might be the most imperceptible crack, or it might be a vast chasm that we simply have not the mental clarity to see.

Exactly. And the trick is indeed to shrink that semantic cool person as small as possible so as to maximise the syntactical range of mathematical machinery. So it is no surprise that the necessity of semantics remains so well hidden from the casual gaze.

It is a religious attitude - going back to Pythagoras and Plato - that the mathematician is really divining the mind of god. It is pure reason that reveals the fundamental axioms on which all truths are built. Crude and brutish materiality in the form of sensory impressions is kept outside the door of the temple.

Why, part of the mythology of maths is that real mathematicians don't sully their thinking with visual imagery - intuitions are the mark of the female or primitive mind.

Every society must have its creation myths.