Re: Mathematics: Is it really what you think it is?
Le 27-janv.-06, à 10:08, Marc Geddes a écrit : For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be time indexed (time dependent)? Or are all mathematical truths really fixed This is a curious question given that platonism, by definition, is the doctrine that (mathematical truth) is fixed. Now, what perhaps you are missing, is that, even just arithmetical truth (and even less) defines canonical person point of views (hypostases) including some which develop temporal logic and dynamics. This is a consequence of incompleteness. Your question was really is platonism false? All naturalist (physicalist, materialist) believes that platonism is false, since Aristotle. I mean the monist or neoplatonist understanding of Plato. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Mathematics: Is it really what you think it is?
Hi Marc, I share with you a feeling that there is something missing in the static picture of mathematical truth as painted in Platonism; there is no fundamental sense of where Becoming originates. It has been a perpetual problem for Platonistto explain how to derive our sense of change from a fundamental Changelessness, although Bruno et al are making a good case. It seems to me, following ideas like those of Jakko Hintikka with his game theoretic idea in proof theory and Chaitin's Omega, that we might consider that Becoming is fundamental and that the tautologies of mathematical Truths can be considered as "fixed points" within the overall Becoming. Kindest regards, Stephen - Original Message - From: Marc Geddes To: everything-list@eskimo.com Sent: Friday, January 27, 2006 4:08 AM Subject: Mathematics: Is it really what you think it is? Open question here: What is mathematics? ;) A series of intuitions I've been having have started to suggest to me that mathematics may not at all be what we think it is! The idea of 'cognitive closure' (Colin McGinn) looms large here.The human brain is not capable of direct perception of mathematical entities. We cannot 'see' mathematics directly in the same way we 'see' a table for instance. This of course maynot say much about the nature of mathematics, but more about the limitations of the human brain.Suppose then, that somevariantof platonism is true and mathematical entities exist 'out there' and thereis *in principle* a modality (a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super-intelligence 'see' ? Perhaps there's something of enormous importance about the nature of mathematics that every one has over-looked so far, something that would be obvious to the super-intelligence with the mathematical modality? Are we all over-looking some incredible truths here? Again, McGinn's idea of cognitive closure is that the human brain may be 'cognitively closed' with respect to some truths because the physical equipment is not up to the job - like the way a dog cannot learn Chinese for instance. For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be timeindexed (timedependent)?Or are all mathematical truths really fixed? The Platonists says that mathematics under-pins reality, but what is the *relationship* between mathematical, mental (teleological) and physical properties? How do mental (teleological/volitional) and physical properties*emerge* from mathematics?That's what every one is missing and what has not been explained. So... think on my questions.Is there something HUGE we all missing as regards the nature of mathematics? Is mathematics really what you think it is? ;) -- "Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder's eye on the last day"
Re: Mathematics: Is it really what you think it is?
Hi Marc -- it's interesting to wonder about what it would be like to directly perceive mathematics -- but we also have to acknowledge when we ask the question, what are the philosophical assumptions we're smuggling along. For instance, the human brain is not capable of direct perception of tables, either. What raises a flag for me in your question is the following apparent dichotomy: 1) The human brain is not capable of direct perception of mathematical entities 2) We could imagine some super-intelligence that possessed this ability . . . It seems what you're encouraging us to do is this: think about of what it's like when we see a table, and then say to ourselves something like the following sentence: It would be like that, but with *math*. But what makes us think we can imagine this situation coherently? Light from a table excites our photoreceptors in a well- understood way - how could an equation do that? I have always thought it strange how McGinn and others eagerly apply cognitive closure to some of the very areas where we have made recent amazing progress in understanding! In the case of math, what exactly is it that motivates your intuition that there might be something more that we're missing? And is it something that would not apply trivially to any other thing (i.e. - I can look at a rock on the ground, and say to myself, There's something else about this rock that I'm not sensing - but I could imagine a superintelligence who could perceive what I'm missing. My ability to say this sentence to myself doesn't demonstrate anything interesting about the rock.) Best regards Pete On Jan 27, 2006, at 1:08 AM, Marc Geddes wrote: Open question here: What is mathematics? ;) A series of intuitions I've been having have started to suggest to me that mathematics may not at all be what we think it is! The idea of 'cognitive closure' (Colin McGinn) looms large here. The human brain is not capable of direct perception of mathematical entities. We cannot 'see' mathematics directly in the same way we 'see' a table for instance. This of course may not say much about the nature of mathematics, but more about the limitations of the human brain. Suppose then, that some variant of platonism is true and mathematical entities exist 'out there' and there is *in principle* a modality ( a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super- intelligence 'see' ? Perhaps there's something of enormous importance about the nature of mathematics that every one has over-looked so far, something that would be obvious to the super-intelligence with the mathematical modality? Are we all over-looking some incredible truths here? Again, McGinn's idea of cognitive closure is that the human brain may be 'cognitively closed' with respect to some truths because the physical equipment is not up to the job - like the way a dog cannot learn Chinese for instance. For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be time indexed (time dependent)? Or are all mathematical truths really fixed? The Platonists says that mathematics under-pins reality, but what is the *relationship* between mathematical, mental (teleological) and physical properties? How do mental (teleological/volitional) and physical properties *emerge* from mathematics? That's what every one is missing and what has not been explained. So... think on my questions. Is there something HUGE we all missing as regards the nature of mathematics? Is mathematics really what you think it is? ;) -- Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder's eye on the last day smime.p7s Description: S/MIME cryptographic signature
Re: Mathematics: Is it really what you think it is?
On 27/01/2006, at 8:08 PM, Marc Geddes wrote: Open question here: What is mathematics? ;) (SNIP) Suppose then, that some variant of platonism is true and mathematical entities exist 'out there' and there is *in principle* a modality ( a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. This notion sounds almost as outrageous as my recently retracted notion that music is *heard mathematics*. OK then, I'll make one more attempt - could music be the sensory modality by which we perceive mathematical entities? Since Bruno is currently injecting our thinking with a good dose of ancient Greek wisdom could we not also mention Boethius in this context with his harmony of the spheres? Boethius was actually a Roman patrician who was a super Greek scholar of the 6th century. Actually, Pythagoras discovered (or invented) this idea. A part of the cosmology of the Pythagorean is this extraordinary theory of the harmony of the spheres, which caught later generations in the ancient world and the Renaissance. It is generally accepted by scholars that Pythagoras himself was the first to formulate that concept, which reflects the whole cosmic plan and showed the intimate connection between the laws of mathematics and of music. Aristotle characterizes the Pythagorean as having reduced all things to numbers or elements of numbers, and described the whole universe as a Harmonia and a number. Aristotle continued: They said too that the whole universe is constructed according to a musical scale. This is what he means to indicate by the words and that the whole universe is a number, because it is both composed of numbers and organized numerically and musically. For the distances between the bodies revolving round the centre are mathematically proportionate; some move faster and some more slowly; the sound made by the slower bodies in their movement is lower in pitch, and that of the faster is higher; hence these separate notes, corresponding to the ratios of the distances, make the resultant sound concordant. Now number, they said, is the source of this harmony, and so they naturally posited number as the principle on which the heaven and the whole universe depended. Well - it's surely not such a big jump from this notion to the main topic of this list. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super- intelligence 'see' ? Or *hear* - the architecture of reality as a symphonic musical texture with an infinity of voices/instruments all producing/ generating the laws of physics, the immediate revelation of which are the heavenly bodies in their orbits. Music surely is a form of computation - it has an origin in pure thought and an outcome as a hard physical reality in the form of compression sound waves. A super intellect might be able to hear mathematics as music, indeed the very notion of the harmony of the spheres seems to suggest this. (SNIP) For one thing: Are platonic mathematical entities really static and timeless like platonist philosophers say? What if platonic mathematical entities can 'change state' somehow ? What if they're dynamic? You are now describing essentially a musical phenomenon. Music is dynamic, it modulates and self-references and develops from state to state. And what if the *movement* of platonic mathematics entities *are* Qualia (conscious experiences). Are there any mathematical truths which may be time indexed (time dependent)? Music is of course time-dependent. The canvas that music uses *is* time. I see great room for discussion of your 3-dimensional time view vis a vis musical time-frames. For example the Art of Fugue (Bach's method of composition) stipulates 4 quasi-independent voices or melodies which each exist in their own right and are perfectly satisfying on their own, yet somehow create an emergent, holistic, greater sense of unity when all combined ie. played simultaneously. Each of these melodies is largely unaware of the other three yet dovetails its own sense with the sense of the others. This is what creates the sense of *depth* in musical textures. The experience of listening to music correctly (without talking over the top of it or munching on your dinner) gives the impression of a physical object which you nevertheless cannot see. I would suggest that whatever this object is has been described by the musical process and cannot be perceived in any other way. I hope nobody gets upset by my reintroducing this idea. Once more, a bit of a provocation to up the ante of the discussion. Regards to all Kim Jones People often confuse belief in a reality with belief in a physical reality - Bruno Marchal [EMAIL PROTECTED]
Re: Mathematics: Is it really what you think it is?
Marc, Bruno, Russell, Hal, list, First, a general note -- thanks, Hal, for the link to your paper on the Universal Dovetailer. I have gotten busy with practical matters, so I've gone quiet here. I hope to have time to pursue the UD soon. As to a sensory modality for mathematical objects. The senses and related cultivated intuitive faculties are for qualities and relations that are not universal but merely general (i.e., they're not mathematical-type universals but they're not concrete particulars/singulars either). So to speak, the senses etc. are sample takers, they sample and taste the world. The senses and their cultivated forms and also their extensions (instrumental technological), taking samples, lead to inductive generalizations, and the most natural scientific form of this process is in those fields which tend to draw inductive generalizations as conclusions -- statistical theory, inductive areas of cybernetic information theory, and other such fields (I'd argue that such is philosophy's place, too). Mathematics is something else. Its cognitive modality seems to be imagination, or imagination supported and constrained by reason. Edgar Allen Poe: The _highest_ order of the imaginative intellect is always pre-eminently mathematical, and the converse. http://www.eapoe.org/works/essays/a451101.htm first paragraph's, last sentence. It is to be admitted that Poe counted mathematics as calculating, but, on the other hand, he probably vaguely meant more by calculating than many of us probably would. Imagination becomes the road to truth when the mind considers things at a sufficiently universal level. I.e., two dots in my imagination are just as good an instance of two things as any two things outside my imagination. The imagination along with its extensions (e.g., mathematical symbolisms, the imaginative apparatus of set theory, etc.), supported, checked, balanced by reason, produces fantastic bridges, often through chains of equivalences, across gulfs enormously _divergent_ from a sensory viewpoint. It would all be indistinguishably universal but for abstractions (e.g., sets) whereby one can say that some of these universals are more universal than others, some are unique (as solutions to families of problems, etc.), and the world in its wild variegation (of models for mathematics) can be, as it were, re-created. To say that mathematics is real doesn't imply that it consists of sensory qualities or of the concrete singulars cognized in their historical and geographical haecceity (or thisness) by commonsense perception. It does imply that the kind of cognition which leads to mathematical truth is a cognition of a kind of reality, the reality, whatever it is, of which mathematical statements are true. Of course if we say that only singular objects are real, then there's no mathematical reality. But insofar as such objects are _really_ marked by mathematical relationships, mathematics has enough reallyness to count as reality, unless one wants to multiply reality words to keep track of syntactical level. None of this is to say that the senses ( related intuitive faculties) have nothing in common with imagination. Both of them involve capacities to form creative impressions, to expect, to notice, and to remember. Both of them objectify map, both of them judge measure, both of them calculate or interpret, and both of them recognize (dis)confirm. The mathematical imagination continually honors, acknowledges, and recognizes rules variously old and newly discovered of the games or contracts into which it enters soever voluntarily and whimsically. Now I have to count on the subway's being on time -- if only I didn't have to work! Best, Ben Udell - Original Message - From: Marc Geddes To: everything-list@eskimo.com Sent: Friday, January 27, 2006 4:08 AM Subject: Mathematics: Is it really what you think it is? Open question here: What is mathematics? ;) A series of intuitions I've been having have started to suggest to me that mathematics may not at all be what we think it is! The idea of 'cognitive closure' (Colin McGinn) looms large here. The human brain is not capable of direct perception of mathematical entities. We cannot 'see' mathematics directly in the same way we 'see' a table for instance. This of course may not say much about the nature of mathematics, but more about the limitations of the human brain. Suppose then, that some variant of platonism is true and mathematical entities exist 'out there' and there is *in principle* a modality ( a method of sensory perception like hearing, sight, taste) for direct perception of mathematics. We could imagine some super-intelligence that possessed this ability to directly perceive mathematics. What would this super-intelligence 'see' ? Perhaps there's something of enormous importance about the nature of mathematics that every one has over-looked so far, something that
Re: Mathematics: Is it really what you think it is?
Marc, list, The heck with the train. I'll do chores today instead. I should add to that which I said below, in order to respond to Marc's remarks a bit more specifically. Insofar as any sensory form of mathematical objects will have some sort of flavors in whose terms the senses sample the world, it would actually be kind of restrictive to have a sensory modality for mathematical objects per se. The point of mathematics is the transformability, the rationally supported and constrained imaginative metamorphizability, across sensory senselike information modalities as well as across particular concreta. In a sense, we already have a sensory/intuitive modality or two for maths -- the cultivated sense for space(s) and the cultivated sense for symbols. There'd be no point to regarding one or the other as the one true general model the mathematical reality in itself. Mathematicians will tend sooner or later to try to get beyond that set of flavors or hues or etc., that specialized model. I do certainly agree that the human mind is limited such that there are, very likely, intelligences next to which we're canine or much lower than that. At least, it's hard to disbelieve that there could be and that the possibility is there. But the simplest meanings of this in turn are that our imaginations, intellects, senses, and commonsense perceptions are limited, and that all of them require invite cultivation and extensions in mathematical or scientific research -- and in many other things as well. Now, one can easily suppose these cognitive powers to become so increased that they would be rather unlike anything which we have experienced. But I see no reason to suppose that they would _necessarily_ become comparatively more sense-like than imagination-like or commonsense-perception-like or etc. My guess is that a mind so strengthened would have increased freedom to employ all those modalities variously, integratively, etc. It seems likewise to me that the simplest meaning of the ascribing (I don't mean the limiting) of reality to all established subjects of research, in their full range including maths, is the ascribing of capacities to discover learn about reality to cognitive modes in _their_ full range -- rather than some squeezing of all levels of reality into the subject matters of the sensory modalities, out of a narrow interpretation of reality and a somewhat questionable association of sensory modalities with concrete singulars rather than with the qualities flavors in terms of which the modalities sample taste the world. Yet I think that that idea -- sensory faculties for everything -- actually has some foundation to it as well. For instance, the intuitive sense of a thing's meaning or value as, for instance, a symbol of something else, is a kind of sense-doing-the-job-of-intellect. An intuitive sense of a thing's validity, legitimacy, or soundness as a kind of observational p! roxy for something else, is a kind of sense-doing-the-job-of-imagination. But this sort of thing is only to the extent that the full range from mathematicals to flavors, tendencies, kinds of appearances, to concrete individual things/events, can be squeezed in as subject matters of _any_ of those cognitive modes as employed as scientific/mathematical roads to truth -- (Level IV) imagination (universals), (Level III) intellect (universes, total populations, etc.), (Level II) sensory related intuitive faculties (flavors, non-universal generals, qualities, etc.), and (Level I) commonsense perception (singulars embedded in their concrete historical tapestry -- singulars not as constituting a universe or gamut such that it is supposed that nothing else exists -- instead, singulars among more singulars). For my part, I doubt that platonic entities undergo real change, but they're so rich that they might as well change -- finite minds like ours will never exhaust them, or at least I tend to suppose not. Anyway FWIW that's my story and I've been sticking to it, so far. Best, Ben Udell - Original Message - From: Benjamin Udell [EMAIL PROTECTED] To: everything-list@eskimo.com Sent: Friday, January 27, 2006 8:17 AM Subject: Re: Mathematics: Is it really what you think it is? Marc, Bruno, Russell, Hal, list, First, a general note -- thanks, Hal, for the link to your paper on the Universal Dovetailer. I have gotten busy with practical matters, so I've gone quiet here. I hope to have time to pursue the UD soon. As to a sensory modality for mathematical objects. The senses and related cultivated intuitive faculties are for qualities and relations that are not universal but merely general (i.e., they're not mathematical-type universals but they're not concrete particulars/singulars either). So to speak, the senses etc. are sample takers, they sample and taste the world. The senses and their cultivated forms and also their extensions (instrumental technological), taking
