Does mathematical reality lie outside us ?

[oldman]

Does "mathematical reality lie outside us"?

The mathematician G.H. Hardy 68 years ago expressed his belief that "mathematical reality lies outside us" in "[URL A mathematicians apology, p.35 [/URL]. I'd like to know how philosophers of the new Millennium have decided this question -- if at all. I'd be grateful for comments and/or web references to accepted views.

Here is the context in which Hardy wrote of his belief:

For me, and I suppose for most mathematicians,
there is another reality, which I will call ‘mathematical reality’;
and there is no sort of agreement about the nature of mathematical
reality among either mathematicians or philosophers. Some
hold that it is ‘mental’ and that in some sense we construct it,
others that it is outside and independent of us...

I will state my own position dogmatically ...
I believe that mathematical reality lies outside us, that our function is to
discover or observe it, and that the theorems which we prove, and
which we describe grandiloquently as our ‘creations’, are simply
our notes of our observations. This view has been held, in one
form or another, by many philosophers of high reputation from
Plato onwards...

 

[Fra]

I supposed a pure matematician holds a different view than someone interested in natural sciences. To me mathematics, physics have a symbiotic relation, and this is the perspective I have.

The mathematician G.H. Hardy 68 years ago expressed his belief that "mathematical reality lies outside us" in "[URL A mathematicians apology, p.35 [/URL]. I'd like to know how philosophers of the new Millennium have decided this question -- if at all. I'd be grateful for comments and/or web references to accepted views.

This is an interesting question, in particular since a lot of theoretical physicists also seem somewhat guided by what is called "simplicity" or "mathematical beauty".

Perhaps you've seen this but an interesting reflection is
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
-- http://nedwww.ipac.caltech.edu/level5/March02/Wigner/Wigner.html

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.

To the above, breaths a realism view of things that doesn't go well with my personal philosophy. In a certain sense one can ask what the difference between invention and discovery is, and I think one difference is when one uses a hidden structure implied by realism views.

I think our own evolution and the evolution of logic and languages are inseperable. In this sense I personally see mathematics as part of nature and ourselves. I do not choose to think of the world of mathematics as existing objectively independent of us, just waiting for someone to find it.

I think this could have implications for the way we think of mathematics in physics, continuum vs discreteness for example.

The usual argument against this subjective view is that, how can human existence possibly change the value of pi? Well, if the question is asked like that, it obviously doesn't. I think the connection is deepter. It doesn't rest on human level, it goes IMO down to the makeup of our physical world. Mathematics as we humans know it, I think of as a human abstraction of things that exists without us, however, it doesn't exists in the void. I think there is still a physical basis for all abstractions.

I think logic, mathematics and fundamental physics may have a close connection that we yet haven't found.

So if "I believe that mathematical reality lies outside us" refers to humans, I agree that the abstractions exist without us. Though without humans there would be no libraries with math books :) But I don't think it makes sense to think that mathematics exists in a meaningful way outside the physical universe, ie without matter at all.

So I think the philosophy of mathematics to decent extent overlap with the philosophy of physics and knowledge, since mathematics is usually an quantifiable abstraction of their structure.

/Fredrik

 

[oldman]

To me mathematics, physics have a symbiotic relation, and this is the perspective I have...a lot of theoretical physicists also seem somewhat guided by what is called "simplicity" or "mathematical beauty".

I agree, but think that being so guided is necessary, but not always sufficient. Simple solutions to complex physics problems are often very appealing --- but all too often they're wrong.

Thanks for the Wigner reference. Yes, I've read this article, but think that he was making much ado about very little.

Like you, I don't go along with what Hardy says about mathematical reality lying outside us. But he was a very influential mathematician and his long essay (whose URL is in the OP) is a total delight to read. I suspect that his influence encourages the mathematics community in according "mathematical reality" a similar status to "physical reality", which is what all the mathematicians I've come across now seem to believe.

I think our own evolution and the evolution of logic and languages are inseperable...

Here you touch on something whose importance seems to be ignored by modern philosophers of science --- as well as physicists and mathematicians. These folk never seem to take into account the fact that influential people like Plato, Kant, Newton --- even Einstein, Hardy and Karl Popper --- were handicapped by living before our place in the scheme of hominid evolution was properly understood. As a result the philosophy of science now does not seem to recognise the importance of the evolution of language (especially mathematics) in elevating our status to its present precarious height. The bearing this has on our ideas of what kind of subjects mathematics and physics in factare interests me.

I'm a bit puzzled when you say:

I agree that the abstractions exist without us.

But if we weren't here to do the abstracting? What then?

 

[Fra]

I agree, but think that being so guided is necessary, but not always sufficient. Simple solutions to complex physics problems are often very appealing --- but all too often they're wrong.

I agree it's not always right. The problem is in part exactly what this simplicity is. Clearly simpliticty is a matter of point of view, or a relative notion depending on your premises. Many people seem to have tried to solve this "problem of simplicity". Popper has a chapter called "the problem of simplicity" in his book logic of SD, and his opinion there is that the kind of simplicity called "mathematical beauty" is not good enough because it's ambigous. Instead he suggest a measure of simplicity in terms of what he calles degree of falsifiability. A simple theory is one which is "easily falsifiable".

However I disagree with a lot of Poppers reason for the reason that he puts insufficient focus on progressive evolution.

I think the idea of simplicity that goes under many names is interesting, but it is rarely characterized well, and I think it is also unavoidably a relative concept. There is I think no clear universal definition of simplicit, but that doesn't remove the principle it's value.

I'm a bit puzzled when you say: But if we weren't here to do the abstracting? What then?

I was arguing both in favour of and against Hardy, and i meant that kind of at two different levels. Ultimately, one can say that anything *I* think or know is relative to me, and if I am not here to support the view, then it doesn't exist. I think this is correct. In this sense I disagree with hardy.

But still, when we describe the mathematics of inanimate matter, that MODEL is of course also still supported the human existense, but such a point seems too obvious to be the final comment, so I was suggesting that while matter does not make books, and use computers and symbolic logic printed on papper, matter might itself be a manifestation of a kind of representation, that could sort of be isomorphic to human mathematics. In this subtle sense, I think Hardy has a point.

/Fredrik

 

[mal4mac]

Penrose adopts a similar position to Hardy in Road to Reality (you just need to read the first few dozen pages to see his Platonic position!) Lakoff & Nunez hod that mathematics is 'embodied' and has no Platonic reality.

"the reality of a mathematical concept has more to do with what it does than with what it is." - Timothy Gowers

http://www.amazon.com/gp/product/0192853619/?tag=pfamazon01-20

 

[oldman]

...The problem is in part exactly what this simplicity is. Popper...suggest a measure of simplicity in terms of what he calles degree of falsifiability. A simple theory is one which is "easily falsifiable"...I disagree with a lot of Poppers reason for the reason that he puts insufficient focus on progressive evolution.

Thanks for explaining Popper's view. I agree with you that progressive evolution can help to validate a simple theory also. Perhaps an example is cosmology, where the simplest (flat) FRW version of an expanding model universe has survived down the years to become the simple foundation of the modern consensus, embellished as it is with strange concepts such as inflation, dark matter and dark energy. Would you call this 'progressive evolution'?

Penrose adopts a similar position to Hardy in Road to Reality (you just need to read the first few dozen pages to see his Platonic position!) Lakoff & Nunez hod that mathematics is 'embodied' and has no Platonic reality.

Yes, one can easily gather that Penrose is a closet Platonist. But I'd not heard of Lakoff and Nunez's contrary views. Thanks. Do you have a reference for this?

 

[Fra]

Perhaps me calling it progressive evolution isn't the best choice, but what I meant is to differetiate with two different ways of measuring change: absolute or relative. I think that due to evolution itself, the original premise is lost. History lives on in the present, it is not retained unambigously, like storing time hisory on an infinite tape.

I do not in general have much faith in the notion of absolute, universal and observer independent measures except in an emergent sense, where I think the limiting procedure can't be bypassed by directly jumping to the limit. I think the best abstraction to be used in physics is that of observer relative measures. This does not prevent however, that a local group of communicating observer may reach an agreement, a kind of local observer indepent measure.

With progressive evolution I meant to prefer the abstraction that evolution is relative, and that I think the premises of evolution "move along" with evolution itself. Therefore mesures, including those of "simplicity" are relative to any state of evolution. So simplicity can be attained in at two ways: evolution of matter subject to measure, or evolution of the measures themselves. And I think that both occurs simultaneously. This is somewhat analogous to GR, where we have changes relative spacetime, and changes of spacetime itself. (This analogy is itself suggestive, but in deep need for further investigation - it's far from clear)

Poppers view of evolving knowledge is seemingly motivated by a dissatisfaction by the inductivism much softer evolution. Popper tries to make the inductive process of learning, a deductive process. The deductive step is the falsification of a theory. Only falsifiable theories should be considered, and theories are corroborated until falsified.

If falsified, the problem becomes that of generating a new hypothesis. Popper doesn't pay this step enough attention IMO. He even hints that this question is one of psychology of a scientist and thus not a question for natural science. I think he escapes too easily here! I think we can do better. I think that his obsession to at all cost stay away from induction strips his strategy from effiency. In my world, the method of hypothesis generation, is also evolving, and the feedback between observation to knowledge is more complex than in poppers abstraction.

I think all changes in nature are ultimately best described by this type of progressive evolution. And to a certain exten GR is of this type, but to some extent it is not. In what I seek the theory itself must live the system itself, and beeing part of the whole we get information constraints.

In GR we consider a curved manifold, but the interesting question is the view of something who is part of this manifold. Just consider all the "map data" concentrated to a point? There are observational aspect of classical GR that is unsatisfactory and I think will remain so at least until we have unification of GR and QM.

Perhaps one problem can even be said to be here, they way we apply mathematics in physics. Mathematically the concept of a limit is fairly clear, but in physics this is less clear. Often limiting scenarious provides effective references. I think this is suspect and it does connect parts of foundational physics to the philosophy of mathematics.

Needless to say, this is all part of my personal philosohpy to which others may disagree.

/Fredrik

 

[oldman]

I think the best abstraction to be used in physics is that of observer relative measures. ...With progressive evolution I meant to prefer the abstraction that evolution is relative, and that I think the premises of evolution "move along" with evolution itself. Therefore mesures, including those of "simplicity" are relative to any state of evolution. So simplicity can be attained in at two ways: evolution of matter subject to measure, or evolution of the measures themselves. ...
/Fredrik

Thanks for setting out your perspective, Fredrik. We've had coms. failures here, so I'm late in acknowledging your help. I'm no philosopher, but I must agree with you about evolution being relative, perhaps in the sense described by Omar Khayyam:

The Moving Finger writes; and having writ, Moves on: ...

Evolution moves on from what now is, and in this sense is relative. I also agree that all change is relarive. And mathematics is the language we use to describe change; easy when it's linear (linear algebra, group theory) but a bit beyond us when it isn't.

 

[Fra]

Thanks for setting out your perspective, Fredrik. We've had coms. failures here, so I'm late in acknowledging your help. I'm no philosopher, but I must agree with you about evolution being relative, perhaps in the sense described by Omar Khayyam:

The Moving Finger writes; and having writ, Moves on: ...

Nice quote, I never heard that before.

I don't see myself primarily a philsopher either but I do acknowledge what is usually classified "philsophical issues" are relevant to progression of science.

Evolution moves on from what now is, and in this sense is relative. I also agree that all change is relarive. And mathematics is the language we use to describe change; easy when it's linear (linear algebra, group theory) but a bit beyond us when it isn't.

Right! This is what motivates me for considering some new mathematical construction of these relative measures. Now if we agree that change is relative, then it should mean that the measure of change is relative. Meaning that the present, somehow implicitly, holds a natural measure of change? Can exploring this be fruitful?

Take the concept of entropy, it is usually relative to the choice of microstructure, and if the microstructure is absolute then so is the measure. If we instead consider something like relative entropy, then the current state defines the microstructure, and thus induces a measure of the space of differential perturbations. But as soon as we have a definite change as oppose to infinitesimal, the measure itself moves along, but only slowly, because the measure might be defined statistically, so the measure is updated with a kind of inertia.

I think this suggest a new logic and new statistics where kinematics is naturally contained. The difference between entropy and action are suddenly gone. Ie. we don't have evolution relative to a gigantic microstructure, and work towards heath death, we have evolution relative to the past.

But an observer is not a single measure, it might be better be seen as system of related measures. And in think when the relations here are worked out, we will see how the trivial heat-flow-like dynamics one might expect from entropy dynamics, will turn out to be more complex. The relation between various measures, of space-time, and other internal degrees of freedom should hopefully reveal the simlicity. I think the simlpicity is that of evolution. Simplicity evolves to amazing complexity, by a relatively simple logic.

I think the logic of complexity lies in it's construction rules, and here I think perhaps physicists can try to argue more realistically of what physics should deal with - a new theory of measurement - but where the measurements are not trivialized into postulated linear operators defined on the space of possibilities. This reconstruction should I think also reconstruct mathematics from the view of physics. Maybe physical mathematics, instead of mathematical physics. In this some key point that one can't avoid are the question of the justification of the continuuum.

This gets very philsophical but still there is a possible new mathematical formalism lurking here, that could or could not, solve a lot of problems. I try to let physics suggest what mathematics is needed, rather that what I think is more common - look into your favourite mathematics (say geometry) and try to see what "possible physics" it can make. I so far I have no clear idea if some existing mathematics will do the trick, or if a new branch is needed. So far my own thinking of this smells more like a complex computer algorithm, rather than a simple closed formulat. At best the induction steps could be simple (I'm sure they will).

/Fredrik

 

[oldman]

You are dealing with deep questions here, Fredrik, which I'm not competent to comment on. But your idea about the relativity of entropy I find interesting. As you say, calculations of entropy depend on microstructure, as in the counting of states or complexions of a particular system.

But then it is a puzzle why the entropy of the consensus early model universe can be said to be "low", when it is thought to be a uniform hot gas/plasma --- a state with a maximum number of complexions, as we normally count them. This does puzzle Roger Penrose, as well as humble folk like myself. Penrose argues that it's due to the un-activated gravitational complexions of the early universe. Maybe he's right. But it could be time to consider entropy as a relative thermodynamic property --- in the sense you suggest.

I just have no idea of how this could be done.

Perhaps you will devise a way along the lines you outline here?

...there is a possible new mathematical formalism lurking here, that could or could not, solve a lot of problems. I try to let physics suggest what mathematics is needed, rather that what I think is more common - look into your favourite mathematics (say geometry) and try to see what "possible physics" it can make. I so far I have no clear idea if some existing mathematics will do the trick, or if a new branch is needed. So far my own thinking of this smells more like a complex computer algorithm, rather than a simple closed formulat. At best the induction steps could be simple (I'm sure they will).

/Fredrik

 

[coberst]

The mathematician G.H. Hardy 68 years ago expressed his belief that "mathematical reality lies outside us" in "[URL A mathematicians apology, p.35 [/URL]. I'd like to know how philosophers of the new Millennium have decided this question -- if at all. I'd be grateful for comments and/or web references to accepted views.

Here is the context in which Hardy wrote of his belief:

I am a retired engineer which means that I have studied lots of math and did learn in my college education how to do math but was never taught how to understand math. In the last ten years I have developed an understanding of math, at least I think that I have.

My understanding of math leads me to conclude that G.H. Hardy is full of prunes.

I say that math is a science of pattern. Math is useful in the natural sciences because nature displays a pattern of structure. Wherever there is pattern then math can be useful.

In an attempt to make the new and revolutionary theories of cognitive science, as defined in “Philosophy in the Flesh” by Lakoff and Johnson, clear I will attempt to illustrate how this human characteristic, claimed by the theory, is used in developing arithmetic. I am not saying that conceptual metaphor theory was available when arithmetic was first ‘invented’ I am saying that the book “Where Mathematics Comes From” written by Lakoff and Nunez illustrates how mathematics can be explained using conceptual metaphor theory. In other words because we are metaphorizing creatures we are able to create mathematics.

At birth an infant has a minimal innate arithmetic ability. This ability to add and subtract small numbers is called subitizing. (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol). Many animals display this subitizing ability.

In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.

Subitizing ability is limited to quantities 1 to 4. As a child grows s/he learns to count beyond 4 objects. This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).

“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”

“Conceptual-blending capacity. You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”

We commonly think of metaphor as something like analogy. We are trying to explain something to someone and we say this something new is very much like this other something you are familiar with. This is one form of metaphor but there is another metaphor that is automatic and unconscious. The child playing with objects has an experience of collecting objects in a pile. This experience results in a neurological network that we might identify as grouping. This neurological structure that contains some sort of logic related to this activity serves as a primary metaphor.

The child has various experiences resulting from playing with objects. These experiences result in mental spaces with neural structures that contain the logic resulting from the experience. When the child then begins to count perhaps on her fingers these mental spaces containing the experiences automatically map to a new mental space and become the logic and inference patterns to make it possible for the child to count because counting contains similar operations.

Primary metaphors are the contents of mental spaces developed in experience and the contents then pass to another mental space to become the bases for a new concept. The contents of space A is mapped to space B to then be the foundation for the new concept at space B. This mapping is automatic and unconscious. This mapping of content from one mental space to another is conceptual metaphor and it is the source of abstract ideas.

Quotes from “Where Mathematics Comes From” by Lakoff and Nunez

 

[oldman]

...I have studied lots of math and did learn in my college education how to do math but was never taught how to understand math.

I'm in a similar position, Coberst, except that I'm merely a retired physicist! At a late stage I've become interested in what mathematics 'is'. But I've never learned much about philosophy, so I'm a bit handicapped.

Thanks for your reply and the quotes from Lakoff and Nunez (to whom I'd also been referred in a previous post by Mal4mac). From what you say it seems that mathematics starts off in a primitive way as an innate skill that we are born with. Some folk must be born with more of it than others, as with musical ability! It seems that mathematical potential could be an evolved hard-wired skill like our need to learn and speak a language -- any language.

I'm rather astonished by the way that nearly all mathematicians seem to believe, like Hardy, or at least suspect, that their subject is an eternal truth existing somewhere 'out there'. I guess on planet Plato. It has been a nice surprise to find that there are people like Lakoff and Nunez and yourself who think otherwise.

 

[coberst]

I'm in a similar position, Coberst, except that I'm merely a retired physicist! At a late stage I've become interested in what mathematics 'is'. But I've never learned much about philosophy, so I'm a bit handicapped.

Thanks for your reply and the quotes from Lakoff and Nunez (to whom I'd also been referred in a previous post by Mal4mac). From what you say it seems that mathematics starts off in a primitive way as an innate skill that we are born with. Some folk must be born with more of it than others, as with musical ability! It seems that mathematical potential could be an evolved hard-wired skill like our need to learn and speak a language -- any language.

I'm rather astonished by the way that nearly all mathematicians seem to believe, like Hardy, or at least suspect, that their subject is an eternal truth existing somewhere 'out there'. I guess on planet Plato. It has been a nice surprise to find that there are people like Lakoff and Nunez and yourself who think otherwise.

I suspect that the academic leaders in the field of math just like the academic leaders of the field of philosophy are inclined to give their subject matter a degree of enchantment. They either fail to understand their own specialty or they understand it but wish that the public place their specialty on a high pedestal by thinking it is more magical than it is.

 

[Fra]

But your idea about the relativity of entropy I find interesting. As you say, calculations of entropy depend on microstructure, as in the counting of states or complexions of a particular system.

But then it is a puzzle why the entropy of the consensus early model universe can be said to be "low", when it is thought to be a uniform hot gas/plasma --- a state with a maximum number of complexions, as we normally count them. This does puzzle Roger Penrose, as well as humble folk like myself. Penrose argues that it's due to the un-activated gravitational complexions of the early universe. Maybe he's right. But it could be time to consider entropy as a relative thermodynamic property --- in the sense you suggest.

I just have no idea of how this could be done.

Perhaps you will devise a way along the lines you outline here?

Although I skimmed Penrose thinking of his gravitationally induced collapse of superposition and other things not too long ago I don't have on top of my head his exact reasoning on the entropy matter, but here is briefly how I think of this.

The mental picture of a large external phase of configuration space, in which the universe as as whole evolves is exactly they way of thinking I do NOT think is intrisic. One reasons that since we expect the second law to hold, it seems the state of the big bang or generally "a remote past" have very low entropy, and thus qualifies as an a priori extremely unlikely initial conditions to be explained by conicidence.

If you instead take the intrisic nature of measures seriously, one must ask what is the basis for the fictive configuration space of the universe or possible universes is, at the time of big bang?

My idea is that evolution of observers and evolution of measures are associated, and the point is then that at some imagine big bang, there were no complex observers, the only observers around were themselves random fluctuations. Which means that the intrisic relative measure that can have physical significance is those relative to these primitive observer complexes. And this clearly bounds the entropy.

So my first suggestion to resolve the paradox is to note that the physical measures existing of of entropy, is different at the big bang. I here distinguish between the measures s, and the measurement s(x). We are comparing s(x_now) with s'(x_big_bang) and they are not the same.

I thus have a dynamical view of the concept of degrees of freedom. An unactivated degree of freedom as you say, is IMO not distinguishable, and therefore is no degree of freedom. Only the, to the observer, distinguishable complexions are valid to build physical measures on.

So I think lage part of the problem is what we humans thinking up references and then try to project the origin of the universe on the references of today. I think this is quite simplistic and not consistent with what I personally think is a better way.

Ultimately, the usual notion of entropy might not be unique anway. If we consider what the purpose of entropy is, it's a measure to rate disorder or missing information. And what is information? IMO, I think information is relative too. Meaning that any state of information induces a natural measure on any differing position. This serves as a guide. And when you go from simple measure, to systems of measures, this relative entropy or information divergence (for example kullback liebler) is a measure of how much information need to change in order to go between two states, and this is asymmetrical ie s(a,b) is not s(b,a) so we have an built in arrow of time. And in the complex case which I'm still working on, this naturally goes from an entropy concept to an action concept, and quantisation will (I think) appear as a result of internal transformations in the observer complex, subject to the constrain of conversving the information capacity. In this constructions I am quite sure that an inertial concept will emerge automatically.

In the end I expec this analysis to provide an explanation of the whole idea of actions and path integrals and the evolution of measures (form of the lagrangians etc) by a unified logic. This should probably also imply a GUT.

But needless to say there is a long way there. But my basic starting point and guide, from this "ought to" follow, is the idea of evolving intrisic measures, and the disagreement between different measures are the reason for interactions.

I find that it's necessary to start from such a basic reconstruction because most standard treatments starts at some semiclassical level with a massive baggage, and this really blurs the logic as I see it.

/Fredrik

 

[Fra]

With GUT I mean a FAPP-type GUT. I do not believe a universally fixed theory, the GUT would probably more be a type of GUT of the scientific method, or theory of evolution, rather than "theory of matters", but at any state of evolution a FAPP-type of GUT will exist.

/Fredrik

 

[oldman]

I suspect that the academic leaders in the field of math just like the academic leaders of the field of philosophy are inclined to give their subject matter a degree of enchantment. They either fail to understand their own specialty or they understand it but wish that the public place their specialty on a high pedestal by thinking it is more magical than it is.

Which is to say that folk immersed deeply in most fields of human endeavour have human, frailties, like you and I. Or perhaps we're exceptions that prove the rule! Thanks for the comment.

 

[oldman]

...there is a long way there. But my basic starting point and guide, from this "ought to" follow, is the idea of evolving intrisic measures, and the disagreement between different measures are the reason for interactions.

I find that it's necessary to start from such a basic reconstruction because most standard treatments starts at some semiclassical level with a massive baggage, and this really blurs the logic as I see it.

A long and interesting pair of posts. Thanks, Fredrik. Let me inject a note of simplicity, though --- with respect to your idea of 'evolving measures' that runs through much of what you say.

Before getting on with GUT's and "actions and path integrals and the evolution of measures (form of the lagrangians etc) by a unified logic", consider something much more simple and ordinary --- the concept of number --- a basic measure that underlies the whole of mathematics and allows us to describe quantitatively the world we find ourselves to be in .

Consider first the nature of the simplest kind of numbers, the counting numbers, a subset of what we now call the infinite set of 'reals'. Do you think that this concept can also evolve? It's been with us now for a long time, perhaps even in prehistoric times. And we have found it to be a lasting ultimate foundation for our descriptions of past occurences, going way back to quite adequate descriptions of distant corners of the universe as they were billions of years ago.

How can we apply the number concept successfully to such ancient happenings if evolves with the universe?

I feel shifting sands beneath my feet when I think of such evolution!

 

[Fra]

Before getting on with GUT's and "actions and path integrals and the evolution of measures (form of the lagrangians etc) by a unified logic", consider something much more simple and ordinary --- the concept of number --- a basic measure that underlies the whole of mathematics and allows us to describe quantitatively the world we find ourselves to be in .

Consider first the nature of the simplest kind of numbers, the counting numbers, a subset of what we now call the infinite set of 'reals'. Do you think that this concept can also evolve? It's been with us now for a long time, perhaps even in prehistoric times. And we have found it to be a lasting ultimate foundation for our descriptions of past occurences, going way back to quite adequate descriptions of distant corners of the universe as they were billions of years ago.

Oldman, I like your selection of the focus! I also think that you are right that starting to consider the nature of numbers is good. In this way what I picture does overlap some of the philosophy of mathematics. I do think evolution of languges and evolution of what the languages are supposed to express do go hand in hand.

This is what I have tried to do. Before rushing into complex concepts, let's start to look at the simple concepts, and see how they can evolve into complex ones. In particular am I suspect on the quick jumping into continuum measures in physics.

From normal mathematics one way of introducing "real numbers" is by completing the set of rational numbers by the limits of sequences. Mathematically there is no problem in imagining this, but from a physical point of view in the context of the applicability of mathematics to reality, and can have second thouhgts about the meaning of infinite sequences in nature.

They way I do start my own reasoning(anything must start somewhere), is to consider the notion of distinguisability, which in effect defines a boolean state, 0 or 1. Since I want focus on the physics of measurement, I have no realism ideals. The idea is that an observer that isn't totally trivial, should at least be able to distingush two "external events". Next I consider what I call internal states, which is like memory states. By consuming a stream of distiniguishable events, there can develop images in the internal states. I think of the observers thus defined as a system of microstructures, the induces measures, from combinatorical principles. Thus I do not a priori make use of real numbers. Real numbers to me, would be an approximate concept when you consider observers whose complexity is to large that the spectrum of states for all practical purposes are a continuum.

So when I talk about what I personally picture as a reconstruction of measures like entropy and action, I do not think of them as real valued. Their value spaces are combinatorically derived from the underlying concepts. This of course in itself gives a kind of "quantization". In my thinking there is no place for the traditional "quantization" as in - starting from classical physics, and apply some operator substitution trick - I rather hope that this reconstruction would explain WHY these usual tricks work, and on top of that take it one step further.

So I guess what I am suggesting, in parallel to rethinking the physical elements like actions, entropy, spacetime etc, is a reconstruction of a new kind of "continuum" which suggest the "way of taking the limits" does matter, and furthermore that there can be physics going in on the domains where the limits aren't taken. And the association here is that physical limit of information capacity in matter.

If you go from a concept of distinguishability, and jump to a real valued probability, or measure, we are going too fast and introducing degrees of freedom in an uncontrollable way.

Unfortunately I don't have a lot of time for all this since, it's my passion and hobby. I'm trying to work this out in steps. I'm am trying to put all this down in a paper, but I restarted this project of mine less than a couple of years ago, after a long break it will certainly take much more time.

How can we apply the number concept successfully to such ancient happenings if evolves with the universe?

This is a good point which I take seriously. As I tried to suggest above the numbers concepts are living inside the observer complexes, which in the rudimentary form would be "matter". (I do not suggest apperance of anything like boltzmanns braind; the point is of course the evolution goes in steps, no need to go from chaos to a brain in a single stroke of dice.).

I think this constraint, does explain the constrains also on measures. With evolution of measures, I figure that it's contained the evolution of mathematics of these measures. So I do not like the instant adaption of continuum models. It is going too fast and I think we are missing some points along the way.

I did some some bits of thought before I started this reasoning, but onfortunately I couldn't come up with a simpler starting point than the boolean state. But I do not see that as true vs false, I just see it as two distinguishable states. In the case of logic, all there is to it, is state of information or opinion, there is no referencing it as true or false in an absolute way.

Some key questions are to see what happens when the complexity of the observer increases. So in sense I see creation as the creation of observers. (Observers again not meaning biological life, but a more abstraction notion of systems senseing it's environment). One thing is apparent that when the complexity increases, the resolution of distiniguishable states increases, and so does the confidence in states, as there can be an internal statistics in the image, where some observations are repeated and thus forms the relative measures of change in the observers structure. As the complexity further increases, transitions to more complex systems of measures are favoured - this may not be favoured int the low complexity cases because there isn't enough complexity to support the structure from noise.

One things that I think connects to gravity is to understand the mechanics (in my picture here) how an observer-measure-complex can gain complexity(mass??) and I see that this could probably be explained, when the measurements provides feedback to the measure, and as I hope to make some simulations on. two general measures can in general have a attraction, where their complexity takes the place of mass, and "their distance" is defined by relative measures (similary to the information divergence) - the idea is that the two communicating measures both resist and perturb each other, and the net result is that they are slowly evolving towards each other. I do have a naive idea that spacetime and the inertial phenomena is encoded in this eviolutionary logic. But I surely realize that this is proto ideas and that it's my work to realize this, because I am unlikely to convince anyone else to join, at least in this early stage.

/Fredrik

 

[oldman]

...The idea is that an observer ... should ...be able to distingush two "external events" (and) consume... a stream of distiniguishable events...thus defined as a system of microstructures, the induces measures ... numbers (are) concepts ... living inside the observer complexes, which in the rudimentary form would be "matter". ...what happens when the complexity of the observer increases. So in sense I see creation as the creation of observers. ...

I'm getting buried by lots of words between those quoted here, Fredrick, in trying to unravel what you mean. With such a nebulous topic, it helps to give actual examples, like an example of 'a primitive observer who sees only Boolean states' (I like this simplicity). What about an amoeba as this kind of observer? And is an example of an observer who sees a continuum perhaps a mouse, or do you mean something more complicated, like say Zeus? It would help to pepper your discussion with concrete examples of what you are describing, don't you think?

 

[Fra]

I wrote a posted that got lost due to some timeout. i'll write more later but to quickly reproduce what I mean.

I tried to communicating a personal way of analysing the problem (of fundamental physics) at this point, not a complete, finished, testable theory. But no doubt I expect it to grow into one, in the future.

Just to set the associations in place, Amoebae beeing a biological cell are incredibly complex observers! I'm not sure if you were joking or I was so unclear. To be more precise, what I consider to be simple observers are stuff like elementary particles. But still, those we know, like an electron are certainly nowhere near a boolean one. The boolean ones MIGHT be something like the degrees of freedom expected at the Planck domain.

So to be concrete, an electron can be considered an observer. But at this stage I am not able to describe an electron. But as you work your way up in the measure construction, I expect to identify measure complexes with particles. However the problem is that when one observer, describes another observer, the level of irreducibility beceoms relative - which also ultimately means that the "spectrum" if we may so call it of "elementary particles" may be relative to the observer. So this is complicated, and part of the problem. And this means that even this theory I am talking about here, is constrained my MY own limits. This means that you can not expect perfection, I think an understanding of the limits impose by the problem is necessary to understnad it "as good as possible". More later, if this is getting to fuzzy we can drop it here.

My main input in this thread relating to your OT is I guess that I do see some "physics" in the philosophical quest for the nature of mathematics. The rest are really just elaborations on it without ambition to be complete or perfect

/Fredrik