tag:blogger.com,1999:blog-15005476.post1026071586595698386..comments2023-10-10T08:02:18.073-04:00Comments on Rationally Speaking: On mathematical PlatonismUnknownnoreply@blogger.comBlogger84125tag:blogger.com,1999:blog-15005476.post-39733221962024278292013-08-05T15:48:28.649-04:002013-08-05T15:48:28.649-04:00The problem with the claim that "mathematics ...The problem with the claim that "mathematics are abstract representations of the features and behavior of the universe" is that the foundations of mathematics have nothing to do with observations of nature. Mathematical truth is not determined by a relation to observable reality, it's determined by reason and logical rules. These rules are not entirely tautologous (self-evident/true by definition) either, as Godel demonstrated in his Incompleteness Theorem that no system of formal logic can prove every true mathematical statement. In other words, mathematical truth is greater than logic itself.<br /><br />It should also be noted that modern physics requires a rigourous form of calculus to work at all. Calculus isn't dependent on its relation to physical facts, but rather physical facts are determined by using calculus. See the Quine-Putnam indispensability thesis for more on that.Imam Nick Jackson, F.W., L.S.https://www.blogger.com/profile/02008614716375501689noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-48154304915626083752013-06-22T11:49:55.277-04:002013-06-22T11:49:55.277-04:00I'm interested to know if the 3rd statement, o...I'm interested to know if the 3rd statement, of whether mathematics is independent of us, may have any relation to quantum theory of un-limitless possibilities and how our observation (as in the slit experiment) changes things?briandavidleeghttps://www.blogger.com/profile/00357454211780856954noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-18870092980350992692013-06-21T00:10:10.024-04:002013-06-21T00:10:10.024-04:00This comment has been removed by the author.briandavidleeghttps://www.blogger.com/profile/00357454211780856954noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-86774341124852258802012-09-28T20:52:39.773-04:002012-09-28T20:52:39.773-04:00This comment has been removed by the author.Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-43640301503059495482012-09-22T02:38:05.030-04:002012-09-22T02:38:05.030-04:00Filippo, physics is applied math, and pure math ca...Filippo, physics is applied math, and pure math can be abstract, that's the only difference. I'm returning to the reality of physics, which is not affected at all by L & R or mathematical patonism. Space & time under Relativity theory are non-physical (just pure geometry). It's an obvious and necessary point to make that if physical gravitons exist, then Relativity is not pure geometry. <br /><br />You misunderstand the reference, and the relevance of L & R and mathematical platonism. Returning again to reality, it doesn't matter whether gravitons are considered as a cumulative field and photons likewise. Indeterminacy is a consequence of failure in measurement, and its not necessary to consider photons in cumulative groups. Individual photons are emitted and absorbed with specific energies, so your point is an obfuscation. Gravitons? we will wait and see.<br /><br />You, L & R, and Massimo are just obfuscation using defintions and ontology to raise abstracts to reality. Real application is to physicals (physics), and we need accept limitations in measuring physicals. They have common properties measurable in conjunction, including spatial, temporal, interactive, mass-energy, electro., grav., and so on. No need to invent another level of reality for their properties: they are physical properties accurately described by math. Back to reality.Anonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-31112992431900937412012-09-20T00:41:26.310-04:002012-09-20T00:41:26.310-04:00@Marcus
We seem to be confusing math and physics ...@Marcus<br /><br />We seem to be confusing math and physics here: general relativity (GR) is a physical theory and so it is supposed to be more “substantial” than pure mathematics.<br /><br />Also, gravitons are Bosons and so, like all elementary particles, they are indistinguishable. That is, any graviton is indistinguishable from any other. As Ladyman and Ross brilliantly argue in ETMG, indistinguishable particles don't have many of the properties of physical objects envisioned by common-sense realism. A number of gravitons is not so much a plurality of objects as a label denoting a particular quantum state. (The same thing is, of course, true of a certain number of photons.) As Dirac observed, indistinguishable particles don't have definite positions in space at all times. If they did, their positions could be used to distinguish one from the other. Don't be so sure that adding gravitons will make GR any less “Platonic.”<br /><br />Also, indistinguishable particles can't have local hidden variables that determine their otherwise random interactions. (Note that the term “hidden” is a misnomer: the “hidden” variables are supposed to determine the results of experiments and so they are not permanently hidden.) The local not-so-hidden variables could be used to distinguish the particles. This is one of the many reasons why all professional physicists were sure that the Bell inequalities would be violated, even before the Aspect experiment was performed: it was not a question of metaphysics. Of, course the consensus of the professional physicists was totally right. This is one of the facts ETMG is wrong about (they don't affect the general argument, however.)<br />Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-74744788977724571942012-09-19T19:42:24.965-04:002012-09-19T19:42:24.965-04:00@ JP
Hi JP,
I agree with you that math is mostly f...@ JP<br />Hi JP,<br />I agree with you that math is mostly formalism, meaning that it mostly involves deriving results from definite axioms using definite derivation rules. After all, I am an algebraist. I cannot think of math except in the form of strings of symbols - my geometric intuition is close to zero, unfortunately. However, formalism is not what math is about. It is about solving problems in the real world and formalism as a philosophy of math is a total failure in explaining why math is successful in such applications. A materialist form of Platonism seems to be the only explanation of this success: mathematical structures are meta-structures of science - the laws of the laws of nature, so to speak. (Note that the former statement is not a mathematical or scientific statement – it is a metaphysical statement. People that say that they don't believe or ignore metaphysics usually are just professing a naive form of metaphysics: metaphysics is unavoidable for any thinking person.) The relationship of this form of Platonism to structural realism (SR) - not necessarily “ontic” - is obvious.<br /><br />SR and Mathematical Platonism (MP) are not strange and exotic metaphysical systems. They are the implicit “metaphysic” of the “shut up and calculate” physicists and applied mathematicians. Applied mathematicians, in particular, expect that their math will work for every structurally similar problem, independently from the objects being treated. Is this “object independence” that makes math so powerful. Linking the reality of math to particular physical objects would make its power even more of a miracle than it already is. Also, applied mathematicians are expected to produce results in the real world, not in some sort of formal game. I know: math has paid my mortgage, etc. - until I retired from the Los Alamos National Laboratory. Formalism as a metaphysics of math is a nonstarter.<br /><br />Formalism does not explain the progress in mathematics. How does mathematics progresses? The greatest successes of mathematics are not obtained by working carefully within existent systems of axioms. They are not obtained by “playing by the rules.” They are obtained by changing the rules of the game. Think of non-Euclidean geometry: you drop Euclid's fifth axiom, derive new geometries and in a century or so you have general relativity. So, where is the next opportunity for progress in mathematics? My guess is that the next big thing in mathematics will involving changing not only the axioms, but the rules of derivation. In particular, the use of probability methods in the derivation of mathematical “theorems” only probably, but not certainly, valid should significantly extend the reach of useful mathematics.<br /><br />[I have made some corrections and removed the last section that was mostly cut-and-paste from the introduction to one of my crazy papers.]<br />Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-27956335280344915142012-09-19T18:35:24.614-04:002012-09-19T18:35:24.614-04:00Having mentioned space & time (which are funda...Having mentioned space & time (which are fundamentals) I realize that physicality is unclear under Relativity Theory, which attemps to reduce the gravitational (physical) "force" to mathematical geometry. The question arises whether mathematical geometry therefore suffices as an "object" without any physical substantiation, if it is strictly geometrical. <br /><br />Here we are at the edge of practical examples & knowledge. To avoid matematical platonism of some kind here, it requires the physical substantiation of the force of gravitational attraction (by gravitons, for example, rather than as pure space-time geomtery). It is physical, and the geometry would be the most fundamental property of the physical field gravitons and their interactions with massive particles. Realtivity Theory without gravitons is the best hope for mathematical platonism. Anonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-14324042116388117132012-09-19T06:41:27.741-04:002012-09-19T06:41:27.741-04:00Marcus - you are correct - the word real is meanin...Marcus - you are correct - the word real is meaningless - it should be changed to 'real for'DaveShttps://www.blogger.com/profile/15840516954793215700noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-78757735596817719872012-09-19T05:57:36.375-04:002012-09-19T05:57:36.375-04:00So many ism's for a basic issue of abstracting...So many ism's for a basic issue of abstracting by mind and applying it (or not) to reality. Mathematics is real to the extent it applies to describe physical properties (including spatial & temporal properties of physical individuals and their relations) and otherwise not.Anonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-88912643792741261512012-09-18T20:10:03.159-04:002012-09-18T20:10:03.159-04:00@ Filippo
Hi,
I would agree with much of what yo...@ Filippo<br /><br />Hi,<br /><br />I would agree with much of what you say. Mathematicians are certainly not mainly concerned with formalizing their work. I should say perhaps that I, too, have done my share of math. That was some time ago but I did study math for the best part of 10 years, with some undergrad teaching thrown in, stopping just short of a thesis (to work in software). My interests were more into analysis and pure mathematics than yours (from what you mention in your comment). But I can fairly say that I know the trade.<br /><br />This question we're discussing fascinates me and there are obviously many facets to it. The reason I brought formalism in is not because I take formalism as the most important or even significant part of mathematics. No, of course not (although it is always present in the background).<br /><br />I brought it in because I believe formalism provides a very useful point of view in analyzing the predictive power of mathematics. This “power”, I think, expresses itself at least in the following way: we start with some premises (axioms or whatever), apply mathematical reasoning and derive some theorem. Next, we “map” parts of reality (some objects) to the mathematical entities under consideration and expect that the theorem we proved mathematically will also apply to the given objects.<br /><br />Now, here's what formalism does for us. The proof of the theorem in question can be written formally in terms of what are essentially mechanical steps. Then, what seems to happen, is that these formal steps can be transposed and apply to real objects (if they fit the premises of our system well enough). I think this process of transposition of formal manipulations to real objects (back and forth) is the key here.<br /><br /> A simple example. The number 101 is prime – this is our purely mathematical theorem. Now, we (think) we know that, were we to select, anywhere in the universe, 101 small objects (roughly the same size), it would be impossible to distribute them on a plane surface in a (non trivial) rectangular pattern. Why? Because, if we could, this physical manipulation could be transposed back into the formalism of arithmetic and provide a factorization for 101. (Not sure what this says about our universe.)<br /><br />So, you see, we have this equivalence between formal and physical manipulations. My feeling is that, perhaps, more complex situations could be analyzed the same way. This is not to say there is nothing to explain - but, at least, this is a situation we can analyze. What do you think?<br />JPhttps://www.blogger.com/profile/12609837930361362269noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-46410730773117813862012-09-18T14:35:59.575-04:002012-09-18T14:35:59.575-04:00This comment has been removed by the author.Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-17543089348783687352012-09-18T14:30:17.917-04:002012-09-18T14:30:17.917-04:00This comment has been removed by the author.Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-80068847664510348692012-09-18T12:34:22.315-04:002012-09-18T12:34:22.315-04:00Damian,
re: Finally, far from *explaining* it, i...Damian, <br /><br />re: <i>Finally, far from *explaining* it, it seems to me that platonism makes a *mystery of the applicability of mathematics in science.</i><br /><br />Well said. What these so-called mathematical objects are and how we can come to know them is under mathematical platonism a far greater mystery than how a non-mathematical realist might go about explaining the utility of mathematics to the sciences. Cian Eamon Marleyhttps://www.blogger.com/profile/09070168038290681070noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-64612591387192167702012-09-18T01:42:55.141-04:002012-09-18T01:42:55.141-04:00There's also Pat Churchland's suggestion t...There's also Pat Churchland's suggestion that mathematical platonism is *biologically implausible*, given what we know about the kinds of organisms we are and how we obtain information about the world. Not only must the naturalist who is also a platonist explain how mathematicians are supposed to gain access to eternal, causally inert entities or structures, and a nontrivial account of how they are referred to ('nontrivial', that is, in the sense that it cannot amount simply to stipulation, as Shapiro's does), but he or she is also obliged to provide a plausible account of the selective advantage that being able to do so would confer upon our ancestors. To paraphrase Churchland, what would have been the evolutionary pressure for the emergence of a special faculty for discerning immutable rational structures that do not exist in the physical universe? What could have been the nature of such a pressure? Is there a plausible account consistent with natural selection that can explain how humans could come to have such a capacity?<br /><br />Finally, far from *explaining* it, it seems to me that platonism makes a *mystery of the applicability of mathematics in science. As Charles Chihara asks, why would empirical scientists need to discover complex and complicated relationships between entities that do not belong to the physical universe in order to develop their empirical theories about what *does* belong to it? Why is knowledge of these undetectable, nonphysical, causally inert entities needed in order to discover facts about the empirical world?<br /><br />Unless a feasible epistemological account is given of how such ontologically mysterious objects may be accessed and interacted with such that they *constrain* mathematical theorising, along with an account of how the postulation of such objects explains the applicability of mathematics to the physical world, it seems to me that their postulation is explanatorily redundant and thus ought to be eschewed on grounds of ontological parsimony.<br /><br />There are many other objections one could raise to the idea that mathematical platonism is compatible with naturalism, but I’ll leave it at that for now.Damianhttps://www.blogger.com/profile/13924807028382126479noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-56239619142311463962012-09-18T01:41:28.514-04:002012-09-18T01:41:28.514-04:00I've only just seen this and haven't had a...I've only just seen this and haven't had a chance to read all the comments, but I would urge Massimo to be more sceptical of the idea that mathematical platonism (or even anti-nominalism) is compatible with naturalism, not least given the seemingly insuperable epistemological difficulties it gives rise to. <br /><br />In a nutshell, the platonist needs to provide a naturalistically acceptable explanation for the consistent reliability of the correlation between mathematical beliefs and the mathematical facts, but since (ex hypothesi) the latter casually inert, exist outside spacetime, and are wholly mind- and language-independent, it's hard to see how any naturalistic explanation is possible even in principle. <br /><br />Indeed, if one’s philosophy of mathematics is platonist it seems difficult to avoid Mark Steiner’s conclusion that there must be some kind of pre-established harmony between the rational structure of the universe and the a priori cognitive faculties of the human mind. I take it that if mathematical platonism does lead to such a conclusion, this must be taken by the naturalist as a reductio of mathematical platonism. <br /><br />At the very least, a world in which there exists eternal, intelligible mathematical entities and structures knowable purely conceptually suggests a rationalist metaphysics and epistemology which the naturalist ought not to countenance merely on the grounds that the prima facie meaning of mathematical truth statements suggests that there ought to be something to which such statements refer. <br /><br />Stewart Shapiro, the leading proponent of mathematical structuralism (of the so-called "ante-rem" or platonist variety), at least recognises where the burden of proof lies here: <br /><br />"Any faculty that the knower has and can invoke in pursuit of scientific knowledge must involve only natural processes amenable to ordinary scientific scrutiny. The realist thus owes some account of how a physical being located in a physical universe can come to know about abstracta like mathematical objects. […] The burden is on the realist to show how realism in ontology is compatible with naturalized epistemology". <br /><br />Unfortunately, however, Shapiro’s own attempts to sketch such an epistemology by recourse to a just-so story about pattern recognition and abstraction at best provides a hypothetical account of elementary arithmetical *concept-acquisition*. Apart from the fact that cognitive science does not bear out these hypotheses, showing how one might acquire concepts of mathematical objects is obviously not sufficient to establish that the objects to which these concepts purport to refer do in fact *exist*. In response to this objection Shapiro states that when it comes to mathematics "an ability to coherently discuss a structure is evidence that the structure exists". Why this should be the case for mathematics when it is patently not the case for any other area of human discourse Shapiro does not tell. Damianhttps://www.blogger.com/profile/13924807028382126479noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-76930464123185726992012-09-17T20:46:20.527-04:002012-09-17T20:46:20.527-04:00Unfortunately I must avoid much of the detailed re...Unfortunately I must avoid much of the detailed referencing as its mostly unfamiliar to me, but I think the problem boils down to the human capacity to abstract mathematics and also apply it to the world. The fact that the world can be understood mathematically, by definition means that the world is mathematical in its properties so described. The fact that chess players can use mathematics in a less useful way means that they can abstract a use for it for their amusement. <br /><br />The relation between something abstract and its real application is the purview of rationality (continually matching the abstracts that proceed realization at every moment of thinking). The realtion is not concrete, and the issue is whether there are pure abstracts at all, given there will some elements of an abstract that is of use in some way to model reality, even if put togther in a novel way.<br /><br />I would go with an innate capacity in humans to observe, abstract from observation, apply the abstracts to model reality, and to try to model it also for its own sake at times (as games for example). As I suggest, the line of abstract is not concrete and it is an ongoing process of human thought to sort what is extrapolated from observation and fashioned into novelty with greater or less modelling potential in reality. QEDAnonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-5641078814267183292012-09-17T18:47:11.996-04:002012-09-17T18:47:11.996-04:00Hi Filippo,
Thanks for your thoughts.
I think Gö...Hi Filippo,<br /><br />Thanks for your thoughts.<br /><br />I think Gödel's theorem is a problem for the following view: (1) there exists some kind of an external realm of mathematical objects/truths; and (2) we can get at it through a formal system. This is not what I am saying at all, as I find (1) very problematic.<br /><br />And, yes, Gödel was a Platonist. He believed for instance that the continuum hypothesis (although shown to be undecidable using common axioms) is nevertheless true of false in some absolute sense. With all due respect to old Kurt, I must disagree on this one.<br /><br />But, whatever one may think of this elusive realm, fact remains that mathematical theories can be formalized. All the theorems of mathematics ever used in physics are deducible from the axioms, and from a relatively small set at that.<br /><br />And here's the thing: all these theorems found so useful and applicable to the real world <i>can be deduced in a completely formal manner from the axioms</i> – with no reference whatsoever to spooky mathematical objects and all. Take this as an alternate formulation of the so-called “no-miracle” question: how can purely formal deductions, these (mechanical?) symbol manipulations within a formal system, produce these useful results? I don't see how mathematical Platonism helps to solve this.<br /><br />You are right to point out that axioms must be selected and that formalism is no help here. But the problem at hand is the question of predictions and it is the same whatever method we use to select axioms. I also agree all mathematical results are implicit in the axioms (you formulate this in terms of information), the same way all chess games, however brilliant, are implicit in the rules.JPhttps://www.blogger.com/profile/12609837930361362269noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-54826691879767059742012-09-17T16:29:49.005-04:002012-09-17T16:29:49.005-04:00@ Massimo
> A better question is: where *is* i...@ Massimo<br /><br />> <i>A better question is: where *is* it testable? Answer: nowhere.</i> <<br /><br />"<a href="http://www.sheldrake.org/Articles&Papers/papers/morphic/pdf/formative.pdf" rel="nofollow">A experimental test for the hypothesis of formative causation</a>."<br /><br />> <i>No, it doesn't, partly for the simple reason that the concept of morphic field is empty.</i> <<br /><br />I stand by my previous statement. Your "mathematical platonism" presupposes something very much akin to a morphic field - an immaterial realm of abstract forms. (There's no point to debate this issue any further because you will simply continue to deny the parallel - a denial that doesn't even begin to address the argument I just put forth.)<br /><br />> <i>Add one more professional academic to be replaced by Wikipedia wisdom.</i> <<br /><br />You're attempting to defend what is ultimately indefensible. Why there is something rather than nothing is a question that science cannot address. <br /><br />> <i>Congrats for having advanced to at least consider what professionals have to say. But it ought to be clear that Sorell's definition doesn't even apply to people like Ladyman and Ross - who are among the best candidates - so it is hard to imagine how it applies to "philosophy" in general.</i> <<br /><br />To reiterate: Ladyman and Ross begin their book ("<a href="http://www.amazon.com/Every-Thing-Must-Metaphysics-Naturalized/dp/0199573093/ref=sr_1_1?s=books&ie=UTF8&qid=1347913566&sr=1-1&keywords=Everything+Must+Go+Ladyman+Ross" rel="nofollow">Everything Must Go</a>") by defending "scientism." (The first chapter of their book is entitled "In Defense of Scientism.")<br /><br />Alastair F. Paisleyhttps://www.blogger.com/profile/15732723685886383829noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-36934942131580768842012-09-17T16:29:08.763-04:002012-09-17T16:29:08.763-04:00Massimo,
I understand you wish to bring this thre...Massimo,<br /><br />I understand you wish to bring this thread to a close, so I won't expect a response. <br /><br />I am not claiming (nor do I think it wise to claim) that mathematical propositions are not true. Rather, I am claiming we can say that mathematical propositions are true while, strictly speaking, holding to the non-existence of so-called mathematical objects. [Though, I think framing the truth of mathematical propositions in terms of truth conditions is misguided -- we should instead talk about proof conditions.]<br /><br />The bit about fictional objects goes only to show that (P3) is strictly false: We can quantify over objects which we hold not to exist. Of course I agree that there is something disanalogous between the two cases, but I don't need them to be analogous in order to reject (P3). <br />Cian Eamon Marleyhttps://www.blogger.com/profile/09070168038290681070noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-19524427829641869462012-09-17T15:45:45.150-04:002012-09-17T15:45:45.150-04:00Your description of formalism represents a view of...Your description of formalism represents a view of Mathematics that was fashionable one hundred yeas ago. It has now been largely abandoned. There are two different, but related, reasons why this is so. The first reason is the “no miracles” objection. If doing mathematics consists of “manipulating symbols in a formal system (axioms, undefined primitive terms, inference rules)”, then why is the result of such a purely formal process any more useful in the real world than, say, playing chess, which also involves working within well defined, arbitrary rules. If you say that the rules are not arbitrary, after all, then the problem arises of where the rules (axioms and inference rules) of mathematics come from. Formalism cannot provide the answer.<br /><br />The second problem is the discovery of Gödel's incompleteness theorem. Formalism has been devastated by Gödel, Turing and company: it is not a viable option anymore. The simplest explanation of this fact was given by G. J. Chaitin: algorithmic information theory shows that any set of axioms contains way too little information to constrain even a simple(?) field of math like arithmetic. You cannot extract more information out of the axioms than the axioms contain to start with. (Information is conserved.) Because of this, progress in mathematics nowadays involves adding to the rules of mathematics.<br /><br />New axioms or inference methods need to be added in order to avoid getting stuck. Quoting Gödel: '...Perhaps also the apparently unsurmountable difficulties which some other mathematical problems have been presenting for many years, are due to the fact that the necessary axioms have not yet been found. Of course, under these circumstances mathematics may lose a good deal of its “absolute certainty;” but, under the influence of the modern criticism of the foundations, this has already happened to a large extent...'<br /><br />Note that, in the previous quotation, Gödel sounds like a post-modern critic of science (mathematics may lose a good deal of its “absolute certainty”...) In fact, Gödel, arguably the most important mathematician of the 20th century, was an unrepentant Platonist (who are we to disagree...) This is shown in the quote by the use of the word “found” with respect to new axioms. New axioms of mathematics are found out there, like the laws of physics, not arbitrarily defined or fished out of the depths of the human mind.<br /><br />I apologize for adding arguments from real mathematics to this content-free discussion of mathematics...<br /><br />References<br /><br />Gregory J. Chaitin, The Limits of Mathematics, Springer, reprinted in 1998. This book is a model of a modern math book: it is very informal and it is mostly LISP and Mathematica code. You don't have to type the code in yourself, fortunately: it can be downloaded from Wolfram Research (or, at least, it could be last time I checked.)<br /><br />Thomas Tymoczco (Ed.), New Directions in the Philosophy of Mathematics, Princeton University Press, reprinted in 1998.<br /><br />The quote by Gödel is from his contributions to Philosophy of Mathematics, P. Bencerraf and H. Putnam (eds.), Prentice-Hall (1964)<br /><br />Roland Omnes, Converging Realities, Princeton University Press (2005). “...one gains a deeper understanding of the foundations of quantum mechanics when one can rely on the axiom of choice. Said otherwise, this axiom definitely belongs to the language of physical reality, at least as we understand it at present.”<br />Filippo Nerihttps://www.blogger.com/profile/01910861498359320434noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-63343277602307752952012-09-17T09:11:20.134-04:002012-09-17T09:11:20.134-04:00As I said above, it is embedded in the argument fo...<i>As I said above, it is embedded in the argument for existence given at the beginning of the post. </i><br /><br />OK, then you've made a remarkably weak and unconvincing argument. Fine with me! <br /><br /><i>As a mathematician you should appreciate that most math has nothing to do with the physical world and is not developed because of it. </i><br /><br />This is simply historically wrong. From Archimedes to Heaviside to Dirac, much mathematics was developed for its use in physics.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-17893227776370788462012-09-17T08:58:20.338-04:002012-09-17T08:58:20.338-04:00Onedaymore, it could be, or it might not be, I wil...Onedaymore, it could be, or it might not be, I will leave that question to you as it has nothing to do with anything I have said. The best start to answering your specific questions about my book is by reading it, or don't read it and don't start answering those questions, as the case may be. Anonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-44250569165270255362012-09-17T08:46:39.918-04:002012-09-17T08:46:39.918-04:00Sure Dave S, basically I go for predetermination a...Sure Dave S, basically I go for predetermination all the way, which is mega-controversial in the current climate. I lay out an environment for evolution that predetermines structures all the way to humans, and is the product of cosmology all the way from the Big Bang, a big task to make the continuing causal connection all the way. In the process I lay out a fully anatomical basis for the structure of human awareness rather than simply relying on neurology (which the full anatomy uses).<br /><br />In awareness using neurology, I propose humans have the equal capacity for motor anticipation & sensory recognition (immediate future & past), whereas other species favor the sensory in adaptation to niches in which they are embedded. With anticipation, humans can control their actions and not merely react to the environment in a niche - we can plan, test, & stay a ahead of the game in a universal niche. Anonymoushttps://www.blogger.com/profile/14612283941807324298noreply@blogger.comtag:blogger.com,1999:blog-15005476.post-6938423249407283032012-09-17T08:35:37.256-04:002012-09-17T08:35:37.256-04:00Alastair,
thanks for the lesson in scientific met...Alastair,<br /><br />thanks for the lesson in scientific method 101, though I think I knew the difference between theories and hypotheses already.<br /><br />> the question I am actually asking here is: "Why do you believe Sheldrake's hypothesis is not testable?" <<br /><br />A better question is: where *is* it testable? Answer: nowhere.<br /><br />> The point is that your immaterial realm of mathematical abstractions qualifies as s kind of morphic field. <<br /><br />No, it doesn't, partly for the simple reason that the concept of morphic field is empty.<br /><br />> Krauss is invoking "magic" too. <<br /><br />Add one more professional academic to be replaced by Wikipedia wisdom.<br /><br />> "Philosophy in keeping with the new scientism only recognizes the existence of objects that science is already committed to." <<br /><br />Congrats for having advanced to at least consider what professionals have to say. But it ought to be clear that Sorell's definition doesn't even apply to people like Ladyman and Ross - who are among the best candidates - so it is hard to imagine how it applies to "philosophy" in general.<br /><br />Mike,<br /><br />> Mathematics are abstract representations of the features and behavior of the physical universe. <<br /><br />Most mathematics has nothing whatsoever to do with representing the universe.<br /><br />> this kind of stuff reminds me of why I stay away from metaphysics. <<br /><br />Sorry to hear that. Perhaps you will consider reading Ladyman and Ross as an antitdote to your instinctive antipathy for metaphysics.<br /><br />Eamon,<br /><br />> what is the extra-linguistic reality that corresponds to the truth of mathematical statements? I have never found much purchase in postulating objectively existing mathematical objects. <<br /><br />It gives an account of the otherwise miraculous effectiveness of mathematics.<br /><br />> If I do not accept that there are such things as mathematical objects, I will not accept (P1). <<br /><br />Yes, but that would be rather silly, at least if you know anything about math (and I know you do).<br /><br />> It makes perfect sense to quantify over objects that we believe not to exist. Take again the Sherlock Holmes example <<br /><br />You may be confusing existence with mind independence. As you probably know, some philosophers would actually say that Holmes does exist, in the very narrow sense that you define. But I don't buy that, I think that the difference is that in the case of Holmes we have to *stipulate* whether statements about him are true or not, in the case of mathematical truths anyone with sufficient understanding of math can arrive at true statements regardless of any such stipulation.<br /><br />Jeffrey,<br /><br />> Commenters have asked for clarification, but you haven't provided one. <<br /><br />I have. As I said above, it is embedded in the argument for existence given at the beginning of the post. You will find more in Linnebo's original article.<br /><br />> The "unreasonable effectiveness" arguments seem to be missing the mark. We use mathematics to describe physical reality because it is useful and effective; if dancing or architecture were more useful, we might use them instead. <<br /><br />But we don't. You are reversing causal roles here. As a mathematician you should appreciate that most math has nothing to do with the physical world and is not developed because of it. And yet, frequently it happens that very abstract mathematical concepts just happen to match surprisingly nicely with the physical world. Hence the no-miracles argument (which, as I pointed out, works by the same logic as the very convincing one deployed by scientific realists).Anonymoushttps://www.blogger.com/profile/09099460671669064269noreply@blogger.com