Monday, 3 October 2011

The (in)consistency of PA and consensus in mathematics

(Cross-posted at NewAPPS)

Last week, the foundations of mathematics community was shaken by yet another claim of the inconsistency of Peano Arithmetic (PA). This time, it was put forward by Edward Nelson, professor of mathematics in Princeton, who claimed to have found a proof of the inconsistency of PA. A few months ago, quite some stir had been caused when Fields medalist V. Voevodsky seemed to be saying that the consistency of PA was an open question; but Nelson’s claim was much more radical; he claimed to have proved that PA was outright inconsistent! (Here is a great post by Jeff Ketland with a crash-course on PA and a discussion of ways in which it might be inconsistent.)

Nelson announced his results on the FOM mailing list on September 26th 2011, providing links to two formulations of the proof: one in book form and one in short-summary form. Very quickly, a few math-oriented blogs had posts up on the topic; we all wanted to understand the outlines of Nelson’s purported proof, and most of us bet all our money on the possibility that there must be a mistake in the proof. External evidence strongly suggests that PA is consistent, in particular in that so many robust mathematical results would have to be revised if PA were inconsistent (not to mention several proofs of the consistency of arithmetic in alternative systems, such as Gentzen’s proofs -- see here).

Indeed, it did not take long for someone to find an apparent loophole in Nelson’s purported proof, and not just someone: math prodigy and Fields medalist Terence Tao (UCLA), who is considered by many as the most brilliant mathematician currently in activity. The venue in which Tao first formulated his reservations was somewhat original: on the G+ thread opened by John Baez on the topic. (So those who dismiss social networks as a pure waste of time have at least one occurrence of actual top-notch science being done in a social network to worry about!) At the same time, Daniel Tausk, a professor of mathematics at the University of São Paulo, had identified the same mistake in Nelson’s argument, and alerted Nelson of the problem in private communication. Judging from his replies (which can be read in the FOM archive as well as in The n-Category Café post), Nelson did not immediately understand the objection, but within a few days consensus had emerged that the mistake identified by Tao and Tausk was irreparable. Then, on October 1st, Nelson graciously acknowledged the correctness of the objections and withdrew the claim of having a proof of the inconsistency of PA. As has been remarked by many commentators, it is certainly a sign of mathematical greatness to acknowledge one’s own mistakes so promptly!

While it must have taken true mathematical insight to see what was wrong with the purported proof, once the mistake was spotted, it was actually not overly difficult to understand – if not in all its technical details, which presuppose a good understanding of Chaitin’s theorem, at least in its basic idea: it was based on the unwarranted assumption that the hierarchy of sub-theories built for the proof is going to map neatly into the hierarchy of the Kolmogorov complexity of each of these sub-theories. In the game of mathematics, tacitly relying on assumptions is not a good move, even less so when it turns out that the assumption is false.

From the point of view of the social practice of mathematics, this episode raises a number of interesting questions, which can be framed from the point of view of Jody Azzouni’s insightful analysis of how and why mathematics is unique as a social practice. Azzouni argues that mathematics is one of the most homogeneous groups of human practices around, and in particular one which seems particularly favorable for the emergence of consensus. Indeed, mathematicians virtually always agree on whether a purported proof is or is not a valid mathematical proof; it may take a while for enough members of the community to understand its details (as with Deolalikar’s purported proof of $P \not= NP$ last year, but this was mostly because the latter was rather poorly formulated), but past this initial period, consensus virtually always emerges. (To be clear, there can be disagreement on whether a proof is elegant, interesting etc., but not as to its correctness and incorrectness.) This remains a very puzzling feature of mathematical practice, which at least to some people suggests that Platonism is the only plausible philosophical account of mathematical truth.

The swiftness with which consensus emerged as to what was wrong with Nelson’s purported proof is yet another example of the strong tendency towards consensus in mathematical practice – and arguably, it cannot be fully explained as a merely socially imposed kind of consensus, due to homogeneous ‘indoctrination’ by means of mathematical education. That it only took a few days for consensus to emerge naturally also depends crucially on the technologies of information dissemination currently at our disposal – mailing lists, home-pages, blogs, social networks; but more generally, the episode has been a dramatic but classical occurrence of a well known pattern in mathematics as a social practice.

The comparison with another dramatic claim, put forward for the first time more than 80 years ago, may prove to be informative. Indeed, the reception of Gödel’s incompleteness results (which has been extensively studied by Dawson and Mancosu, among others) also illustrates the emergence of a consensus, but this time it favored the highly surprising claim being put forward; in last week’s episode, by contrast, the highly surprising claim has been refuted.

In September 1930, Gödel made the first public announcement of his incompleteness result. In first instance, he had proved only the first incompleteness theorem, and as is well known, from the first theorem the second incompleteness theorem was proved independently by Gödel himself and by John von Neumann. But von Neumann was one of the few who immediately understood the result (Bernays and Carnap seem to have struggled); in fact, he may have been the first to see the far-reaching consequences it had. In particular, while Gödel was still rather cautions regarding the impact of his results for Hilbert’s program, von Neumann immediately drew the right conclusions. However, quite a few mathematicians were so taken aback by Gödel’s results that they were convinced that there must have been a mistake somewhere in the proof; just as in last week’s episode, many of them set out to find out the ‘loophole’ in Gödel’s proof, but on that occasion to no avail. As narrated in this wonderful paper by P. Mancosu, the news of Gödel’s results quickly spread, but in the absence of the WWW or similar platforms, many people did not have immediate access to off-prints of the paper where the results were presented. For example, before seeing the actual results in detail, mathematician Leon Chwistek suspected that Gödel’s proof would be based on misunderstandings, but once he read the actual paper, he stood corrected:

In my last letter I have raised some doubts concerning Dr Gödel’ s work that have completely disappeared after a more attentive study of the problem. I had at first thought that there was a tacit introduction of non-predicative functions which makes the use of a rigorous symbolical procedure impossible and I even feared the possibility of an antinomy. Now I see that this is out of the question. The method is truly wonderful and it fits pure type theory thoroughly. (Letter to Kaufmann, in Mancosu 1999, 44)

Both the consensus that eventually emerged concerning the correctness of Gödel’s surprising results, and the consensus swiftly established as on what was wrong with Nelson’s purported proof last week are examples of the amazing and quite unique tendency to converge towards consensus in mathematical practice. In many senses, the most pressing task for the philosophy of mathematics, practice-based or otherwise, is to formulate a satisfactory account of this almost eerie fact about mathematics.

156 comments:

  1. It would be interesting to also contrast these two episodes with the recent P != NP "proof" that got some press...

    ReplyDelete
  2. @Seamus, it would indeed, but I'm afraid it's not going to be done by me... I haven't followed the P != NP controversy from very close last year, and the topic is too far removed from the areas I feel confident I'll be saying something sensible on.

    ReplyDelete
  3. I suspect that history (ie people) tend to remember episodes where the surprising suggestion turns out to be true better than when the surprising claim turns out to be false.

    ReplyDelete
  4. It's amazing how someone can come along and in a matter of minutes falsify a proof which must have taken Nelson hundreds of man hours to construct.

    And should the original claim have been less bold and more carefully checked before publishing to avoid humiliation and publicity?

    ReplyDelete
  5. Great post! As a mathematician, I find the relative ease of consensus wonderful and mysterious.

    Maybe I'm reading too much into your words, but it seems like you would argue that consensus in mathematics is both easier and swifter than consensus in other sciences like, say, physics. Yet the eventual arrival at consensus is an important feature of physics as well. (For the sake of argument, let's ignore those speculative parts of physics that haven't yet made contact with experiment, and focus on a subfield where theory and experiment see constant interplay, such as condensed matter.)

    It is understandable that consensus in physics would take longer than mathematics: it is an experimental science, and experiments take time. But are there any qualitative differences in consensus in physics, once it is reached, from consensus in mathematics?

    ReplyDelete
  6. "who is considered by many as the most brilliant mathematician currently in activity."

    No serious professional mathematician would accept such a description of any mathematician.

    ReplyDelete
  7. ""who is considered by many as the most brilliant mathematician currently in activity."

    No serious professional mathematician would accept such a description of any mathematician. "

    Funny. I thought Bernoulli believed such a thing about Newton.

    ReplyDelete
  8. With regard to you final paragraph on the "most pressing task for the philosophy of mathematics", I can sympathise with that thought, and yet a huge amount of effort has already been devoted to it. Alongside the justification of formalisms by more orthodox philosophy of mathematics, on the practice-based side we have, say, Ken Manders on the control afforded by Euclidean diagrammatic reasoning in 'The Euclidean Diagram' (1995) and Colin McLarty in Voir Dire in the case of mathematical progress describing the advent of shared means of rigorous expression in the twentieth century.

    To my mind, an equally or more pressing task is to understand the kinds of slowly emerging consensus that occur as to what directions a field of mathematics should take. With the very light constraints afforded by logical correctness or empirical adequacy, there's some explaining to do.

    ReplyDelete
  9. @John,
    Indeed, it seems to me that there is a stronger tendency towards consensus in math than in, say, physics, but that's ultimately an empirical claim of course (the 'sociology' of each of these fields). And it's precisely the fact that math is not an experimental science that makes consensus even more puzzling: there isn't anything like, you know, the physical world, to keep mathematical practice in check!

    ReplyDelete
  10. @David,
    I guess I should have said "the most pressing task for the philosophy of mathematics ... REMAINS to formulate a satisfactory account...". You are absolutely right that there has been a lot of work on the topic, but to my mind it is still essentially an open question.

    The higher level question that you mention, i.e. the directions a field of mathematics should take, is definitely also very interesting, but certainly just as hard or even harder! :)

    ReplyDelete
  11. What exactly is it that you find so mysterious about the strong tendency towards consensus? A naive answer might be that we have a clear notion of what a proof is, and therefore a watertight test that we can apply to claims of mathematical correctness. To the objection that fully detailed and rigorous proofs are never in practice written out, I would answer that mathematicians are trained to know how to take any part of a mathematical write-up and, if it is correct, expand it into something more formal and detailed -- a process that can be iterated if necessary. Experts in different areas get a feel for what is likely to be true in their area, so if an incorrect proof appears, and is of a sufficiently important result, they will usually (but not always) know which part needs to be clarified. Surprisingly often, the mistake is a simple unjustified assertion that the author mistakenly thinks is obvious and a more sceptical reader doesn't find obvious at all.

    I'm curious to have a more detailed formulation of what the real philosophical problem is here. What is unsatisfactory or incomplete about the account (or some modest expansion of it) that I've just given?

    ReplyDelete
  12. What about the objection that the proof itself requires a proof and so on ad infinitum. Why is that objection not justified?

    ReplyDelete
  13. So, accepting Tim Gowers' description of mathematical practice as I do, perhaps there's a question as to why it was possible for the various branches of mathematics to reach the condition he describes. The advent of today's standards of rigour came piecemeal, some branches taking much longer than others. And you can see reactions to the perceived threat of losing the control that it affords in Jaffe and Quinn's concerns about the influx of research by physicists such as Edward Witten.

    That ultimately just about all of mathematics can be rendered in a single and, so far as we can tell, consistent system, ZFC, must tell us something. But the kinds of ways of knowing that there's something fishy in a proposed proof operate at a higher level than that. You can find out quite a lot about this intuition from the Tricki. Category theory also provides a higher-level support.

    ReplyDelete
  14. I sometimes say that philosophy is about 'questioning the obvious'. So while the phenomenon of consensus in mathematics may appear to be fairly unproblematic, especially for those who experience it from within (i.e. professional mathematicians), if you think about mathematics as a body of human practices, I'd say it's quite astonishing. As Azzouni argues in the paper I link to above, in terms of institutionalized human practices, mathematics is quite unique in terms of its tendency towards consensus.

    One can of course argue that this is a product of clear and precise standards, which are transmitted and perpetuated through education. But the comparison with other fields of scientific inquiry can be illuminating here. Take biology, for example: even though (almost) everybody agrees that evolution is the right framework, there is a lot of disagreement on the details: gradalism vs. non-gradalism, adaptationism vs. by-product accounts etc. It's not obvious in which ways education in biology would not be as conducive to consensus as education in math, and yet the situations seem very different.

    And to say that we have a clear, rigorous notion of what a proof is is only the beginning of an answer anyway. As suggested above by David, the emergence of the current standards of what counts as a correct, rigorous proof was a long and winding process. Moreover, while these standards seem widely and unanimously endorsed on the level of practices, i.e. tacitly, a theoretical/philosophical account of what a mathematical proof is is still largely an open question. Typically, a mathematician can tell whether a given purported proof is a 'good proof' or not (as in the case discussed above), but she or he would be hard-pressed to offer a definition of what a mathematical proof is ("I know one when I see one.")

    I realize that these questions may sound pedantic and unnecessary from the point of view of mathematical practice, but that's just philosophy, questioning the obvious...

    ReplyDelete
    Replies
    1. Catarina offers an interesting contrast between the achievement of consensus in mathematics and in empirical science, namely in biology and evolutionary theory. Education in biology and education in mathematics prepares both sets of researchers for speaking the same ‘language’, i.e., mastering the existing concepts and theories, and accepting the forms of argument, reasoning, experiment and proof in the respective discipline. However, at the last hurdle, in evaluating new alternative sub-theories, there are major differences in achieving consensus in the two fields. Why is this? Although there may be minor differences in educational style (mathematics assumes theory consensus, biology perhaps does not) is not the real difference in the disciplines themselves? Biology attempts to account for one world – the empirical world of living things we share. In other words, the truth test is fitting with this one world as best we can (subject to all the constraints indicated by Kuhn). Nevertheless, until theories are falsified or otherwise discarded they remain in contention. Mathematics develops theories based on ideas, concepts, assumptions, problems, etc., and ultimately there is no final competition between different theories. Provided they are consistent they may coexist. We have Euclidean and non-Euclidean geometries. There is no contradiction, no competition, no rivalry unless we apply them as applied theories to the empirical world. And then they are just another part of empirical science. The latter strives for an intended model. Mathematical theories only need to satisfy some particular model, out of the infinitely many possible. So consensus is easier to achieve in mathematics than biology. Which is not to say that the process of proving is easier in mathematics than that of conducting empirical tests in biology. It is just that if your derivation is agreed to be correct there is no further final test of truth, irrespective of the epistemological grounds of that agreement.

      Delete
    2. True. But the puzzling thing is precisely this bit:

      "It is just that if your derivation is agreed to be correct there is no further final test of truth, irrespective of the epistemological grounds of that agreement."

      That is, the consensus on proofs/derivations being correct.

      Delete
  15. As pointed out earlier, consensus in mathematics is a relatively recent phenomenon, basically dating back to the foundational work in the early 20th century. Before then, one had debates about use of infinitesimals, implicit use of the axiom of choice (before this axiom was explicitly formulated), use of complex analytic methods instead of "elementary" methods, etc.

    In my view, one of the main reasons we were able to build such solid foundations to mathematics was that mathematics had been maturing for over 2000 years as a cumulatively developing subject, and that by the early 20th century we had finally reached the point where we could build proper foundations. There are other disciplines that are as old as mathematics (philosophy, for instance), but the cumulative development process started later. (Mathematics, for instance, still uses the classical theorems of Greek geometry, but much of Aristotlean physics or metaphysics has now been discarded by their modern counterparts.) Physics, for instance, has been developing more or less cumulatively since Galileo, and it is conceivable that in a century or two they may have a consensus foundation that is as solid as the foundations of mathematics are today. (They're not quite there yet, though.)

    So perhaps the deeper question is why mathematics started developing in a cumulative fashion much earlier than most other disciplines.

    ReplyDelete
  16. here's a draft mini-paper on convergence in philosophy and why there isn't more of it, by comparison to mathematics, physics, and so on. there's also some data here. of course the foundations of mathematics is intermediate between regular mathematics and philosophy in some respects, so it's not surprising that convergence on the foundations of mathematics is slower than convergence on regular mathematics and faster than convergence in philosophy. it would be nice to pin down the relevant dimensions of difference.

    ReplyDelete
  17. It's important to recognise that advances in mathematics are largely brought about by conceptual innovation rather than by grounding in some or other foundational language. Nineteenth century projective geometry texts may need a little tightening up in terms of rigour from our perspective, but far more importantly we don't need to stand in such awe of the fact that theorems and their proofs can be dualised by the simple expedient of exchanging a few terms, 'line' and 'point', 'passes through' and 'lies on', etc., once we understand the role of the outer automorphism of the group SL(3, R). To understand this we need to understand a range of conceptual advances by Felix Klein and others.

    I could go on and on here. It was a conceptual necessity for Poincaré to introduce early methods of algebraic topology in the 1890s. In doing so he shone a light on earlier work and transformed the mathematics of the twentieth century.

    This isn't to underplay the importance of foundational work, but innovations in the axiomatic method have been as important for allowing conceptual clarification and advance as for the provision of security. How mathematics affords the opportunity for radical changes in its conceptual outlook while preserving a tightly integrated network of theories is the interesting qustion for me.

    ReplyDelete
  18. Good points, @Terry. They clearly illustrate my conviction that the philosophical discussion of these issues must be thoroughly historically-informed if it is to go anywhere.

    I would just like to clarify on what level I identify a tendency for overwhelming consensus in mathematics: it is mostly on the level of recognizing a given proof as correct or incorrect. Of course one can disagree on whether certain assumptions used along the proof are legitimate or not (e.g. the axiom of choice or what have you), but granting the assumptions, in most cases there seems to be consensus as to whether a proof is correct or not. As suggested by David Chalmers above, it is not so surprising that there should be more divergence of opinions in discussions on the foundations of mathematics, where certain assumptions may be questioned or accepted on grounds other than proofs, than in what he calls 'regular mathematics'.

    ReplyDelete
  19. Well, I think the consensus amongst mathematicians is what we created mathematics for. most mathematicians I know have this craving for irrefutable arguments, for always wanting to dispute what has been said by others until no doubt remains (lovely put in cartoon form here http://abstrusegoose.com/353), so in some ways is really not surprising that the consensus is there. sure, it relies on the cumulative nature of the subject. in fact, what is disturbing is when it is *not* there, as is the case of foundations, of the debate between constructive vs classical logic. the need for certainty has turned many mathematicians into constructivists, it seems to me...

    ReplyDelete
  20. William wrote:

    "What about the objection that the proof itself requires a proof and so on ad infinitum. Why is that objection not justified?"

    No mathematicians believe that objection, so even if it were true, it wouldn't affect the way they arrive at a consensus. I.e., we hardly ever mathematicians object to someone's proof claiming the proof needs a proof... although I can imagine someone demanding program verification for a computer-based proof of the 4-color theorem.

    ReplyDelete
  21. "hardly ever" -> "hardly ever see"

    ReplyDelete
  22. I think Valeria hit the nail on the head: mathematicians crave consensus.

    If any sort of argument is of the sort that it only convinces 50% of mathematicians, we'll either say it's "not mathematics", or discuss, polish and/or demolish the argument until convinces either 99% of mathematicians or just 1%. (Example: Cantor's proofs.)

    If someone doesn't play the game according to the usual rules, we'll make up a new game and say they're playing that game instead, thus eliminating potential controversy. (Example: intuitionistic mathematics.)

    Finally, we reward people who quickly admit their errors, instead of fighting on endlessly. We say they're smart, not wimps. (Example: Edward Nelson.) People who fight on endlessly are labelled crackpots and excluded from the community. (Examples: too numerous to list here.)

    ReplyDelete
  23. Well yes, mathematicians crave consensus, but then so do a lot of people. I crave consensus in philosophy (especially if it converges on my views), but I don't expect to find it. The question is why are mathematicians able to satisfy that craving. How have they been able to develop robustly stable practices? How is it possible to sense problems in a proof without dipping down to the machine code of its fully formal rendition.

    It would be fascinating to have a transcript of the thought processes which allowed Terry Tao to see the problem with Nelson's proof.

    ReplyDelete
  24. Great points, @david! I agree entirely.

    ReplyDelete
  25. Dear David,

    Actually, Nelson's proof was relatively easy to understand, in part because he took the trouble to write out a short outline which make clear the general strategy of proof while omitting most of the technical details (though it was ambiguous at one very crucial juncture), and also because I had already previously thought about the surprise examination (or unexpected hanging) paradox and the Kritchman-Raz argument (see the last section of http://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/ ).

    From the outline one could already see that the main idea was to adapt the Kritchman-Raz argument to the theory Q_0^*, which "almost" proved its own consistency in that it contained a hierarchy of theories Q_1, Q_2, Q_3, ..., each of which could prove the consistency of its predecessor.

    Now, I did not at the time fully understand the definition of Q_0^*, nor was I fully aware of the Hilbert-Ackermann result which guaranteed this chain of consistency results, but I was willing to accept the existence of such a hierarchy of theories. (I've since read up a bit on these topics, though.) The question was then, given such an abstract hierarchy, whether one could use the arguments of Chaitin and Kritchman-Raz to establish the inconsistency of at least one of these theories.

    These arguments were simple enough (they were basically formalisations of the Berry paradox and surprise examination paradox respectively) that I could then try to do that directly, without any further assistance from the outline. And, indeed, when I attempted to do this, I did at first seem to obtain a contradiction (much as paradoxes such as the Berry or surprise examination paradoxes also lead to absurdity if one reasons somewhat carelessly using informal naive argument). So I could see where Nelson was coming from; but then I spent some time trying to expand out my arguments in detail to find the error. The key, as I found out, was to specify exactly what proof verifier would be used for the Chaitin portion of the argument (this was an issue that was left ambiguous in Nelson's outline), and in particular whether it would accept proofs of unbounded complexity or not. Since Nelson wanted to keep all proofs at bounded complexity, I used a proof verifier that enforced such a bound, and eventually worked out that this could not be done while keeping the length of the Chaitin machine constant; this was the objection that I raised in my first few comments. However, after Nelson responded, it became clear that he was using an unrestricted proof verifier, and this led to a different problem, namely that the proofs produced by Chaitin's argument were of unbounded complexity. So there was not a single "flaw" in the argument, but rather there were two separate flaws, one of which was relevant to one interpretation of the argument, and the other of which was relevant to an alternate interpretation of the argument.

    Aside from this one ambiguity, though, the outline was quite clear. Certainly there have been other manuscripts claiming major results that were much more difficult to adjudicate because they were written so badly that there were multiple ambiguities or inaccuracies in the exposition, and any high-level perspective on the argument was obscured.

    ReplyDelete
  26. Dear Terry, thanks very much for that. For me it's intriguing how high-level perspectives, phrased in a natural mathematical language, can be sufficiently clear that they allow for the requisite checks to be made. I wonder if in the passage you describe in the second half of the penultimate paragraph, your reasoning is assisted by familiarity with similar potential problems, perhaps arising in very different parts of mathematics.

    In my book I discuss the claims that automated 'analogical' reasoning allowed the production of a proof of Heine-Borel in two dimensions, from a proof of the one-dimensional case. My point was that even in this simple case, the program, which is operating syntactically, requires a large amount of human assistance.

    Put in 'ordinary' language, one would suppose a closed interval with a cover and no finite subcover. Then take half of the interval which can't be finitely covered. Iterate this process. There is a point in the intersection of these nested intervals. It must belong to an element of the cover, and far enough along the sequence the nested intervals must be contained inside that element. Contradiction.

    The one worry for the human trying to extend this to two dimensions is to make sure that one can go far enough along the sequence so that each dimension of the nested rectangles fits inside the given element of the cover. The computer needs a big hand just here.

    Now some truly analogical reasoning would come into play if the human recognises from similar experience that an attempt to work an infinite-dimensional variant of the proof will not work as we cannot be assured that there's an upper bound to distances along the sequence for each of the dimensions to become small enough.



    Would I be right in thinking that

    ReplyDelete
  27. In this specific case, all the mathematicians involved relied for their work on a specific highly developed theory- let me call it a module- and none was prepared to bear the cognitive cost of re-writing the entire module which is why there could be a quick resolution. Surely this happens all the time in other disciplines as well? Indeed the less 'rational' or alethic the subject area the lower the cognitive pay-off for rewriting entire modules so we might find even faster resolution, without even the pause for critical thought. A lesser mind might have taken much longer to concede the point.

    Still, suppose there was a big cognitive pay-off, currently available, for entirely rewriting the Chatin theorem 'module' or rethinking Kolmogorov complexity- it may be that some on or other of those involved might have taken that tack.

    Surely an 'in-built opponent' is a feature of all social communication? It weighs down most heavily where the cognitive cost of re-writing modules are high or the reward is miniscule? I suppose statements about fashion or syntax or what is considered politically correct, have this feature and thus in most social sub-sets there is going to be very quick recantation simply because the cost greatly outweighs any possible benefit.

    At this point, I'd like to introduce a notion of ontologically dysphoria- the feeling of being in the wrong Universe, the intuition that the Cartesian duality of mind and body points to something more troubling, bizarre or tragic. It may be that there is a sort of genotypal canalisation towards a widespread feeling of this sort- perhaps, rather than a malaise attributable to the welgeist, ontological dysphoria is the driver for the necessary-but-not-too-much preference diversity needed to drive trade, but also communication and the elaboration of Knowledge systems.

    It may make a sort of collocational sense for us to agree that Math represents a limit case of one sort and philosophy, with its distinctions without differences, as the limit case of its opposite.

    Godel and Von Neumann, both Theists on their death beds, who agreed on so much in the way of mathematics yet had ontological dysphorias of opposite tropism. It may be that the latter type of 'madness'- or stark solipsistic discontinuity- is as important a driver for breakthroughs in Maths as the great powers of 'reason' both possessed which enabled the Von Neumann to grasp Godel's result, perhaps, more thoroughly and more quickly than he had done himself.

    ReplyDelete
  28. (On rereading this post it seems overly long and maybe boring, especially in its central part "about mathematics". Let's hope someone will be interested. If you aren't, O reader, it's probably my fault rather than yours.)

    As a mathematician, but at a rather low level (retired teacher to junior high school, i.e. pupils 12 to 15 years of age), I believe that the ready obtainability of consensus is a defining characteristic of mathematics:

    Truth in the experimental sciences is defined by agreement with reality, and is always provisional, and subject to the precision of experimental measures.


    Truth in mathematics is defined by internal consistency, and the proof of a really fundamental contradiction (as in the "naïve" axiomatization of set theory, when it was shown that, obviously, "the set of all sets which are not members of themselves" is a member of itself if and only if it isn't) is cause for a serious overhaul of the theory in question (with ZF set theory and its family, but IIUC also with other axiomatizations, all making it so that it is not meaningful to talk about a "set of all sets which are not members of themselves". The solution I'm familiar with is Bourbaki's, who defines the notion of "collectivizability" ("nous disons qu'une relation R est collectivisante en une lettre x si (τ[x](R)|x)R est un théorème (Définition de "collectivisante"). De cette définition il découle immédiatement que la relation x∉x n'est pas collectivisante en x.) (Sorry, my mother language is French and I'm not sure I can provide an exactly faithful translation into mathematicians' English.) Another reformulation (IIUC what is vaguely mentioned about it in Hofstadter's /Gödel, Escher, Bach: an Eternal Golden Braid/) is the theory of types.

    As such, "truth" in mathematics is only relative to a certain choice of axioms: within a given theory (defined by its set of axioms), and if I accept Bourbaki's meaning of "true", then as long as the theory is consistent (which is not always easily provable) there are "true statements" (theorems), "false statements" (whose negation is a theorem), and usually "undecidable statements" (which are neither a theorem nor the contraposite of a theorem). Of course, if the theory is inconsistent, then everything it it is both true and false and _this particular choice of axioms_ loses all interest. Usually, however, for any theory which really interests mathematicians, and always AFAIK for any mathematical theory which interests physicists, engineers, etc., it is possible (not always easily) to find a reformulation of the set of axioms in a way which both avoids the inconsistency just discovered and preserves the already much-used results of the theory.

    Gödel's undecidability theorems establish the surprising (for his time) result that in any theory "powerful enough to develop number theory in it" there is at least one undecidable statement, and therefore an infinity.


    I might be tempted to say that philosophy is characterized by the fact that it is amenable neither to proof by experimentation nor to proof by consistency, and that yet (unlike most variants of theology) it is also not accessible to the argument of authority (i.e. the proof by "Ipse dixit" as the Latins said, or equivalently by "Αυτος εφη" as the Greeks said). But I'm not enough of a philosopher to be sure that this paragraph about the nature of philosophy is valid. (Every mathematician has, of course, to be /some/ of a philosopher, if only to discuss metamathematics and metalogics; but that isn't much as philosophers go.)

    ReplyDelete
  29. Well post its give us good tips and tell us many technique and how to solve question with skills thanks for sharing college readmission letter .

    ReplyDelete
  30. Hi, this is a nice blog thanks for sharing the Informative blog. Myassignmenthelp.com provide the Microsoft case study for a assist to student queries. Our writers conduct thorough research to provide you with a flawless Microsoft case study. My Assignment help brings IKEA pestle analysis .Writing an elaborate IKEA pestle analysis is anything but easy and most students lack the tenacity to work on the details accurately. If you get the more details Please visit the site through Link.

    ReplyDelete
  31. Our company Steven elevator is a reputed name as best Elevator manufacturers in chennai offering an exclusive range of Home Elevators. These elevators are made and designed by a team of expert professionals using optimum grade materials at our state of art machining facility.

    Check the Lift manufacturers in chennai and Elevator companies in chennai

    Follow the steps given at roku.com/link to activate your roku device

    To Setup your brand new HP Printer, Go to 123 hp com setup

    ReplyDelete
  32. A special thanks to My Assignment Help for providing a well-written assignment. You guys were a savior to me. Will contact you soon for my upcoming assignments. You can email us at cs@Myassignmenthelpau.Com or Phone Number: +61-2-8005-8227

    ReplyDelete
  33. For those students who are facing issues while writing their macroeconomics homework, they can give a though to Macroeconomics Assignment Help services.
    Macroeconomics Homework Help

    Business Letter Format

    ReplyDelete
  34. Our team of trusted dissertation writers works on your task with thorough concentration. This results in the completion of work even before the deadline.
    write my essay for me

    write my essay online

    ReplyDelete
  35. Make your assignment submission more effective using assignment help services. This Assignment Help Online service allows students to discuss their concerns with qualified assignment helper and finish their assignment on time.

    ReplyDelete
  36. When your HP scanner stops working, it means that your HP scanner is passing through some technical crisis. If you try to work on your HP scanner, it is not responding properly. So, it may be very difficult situation for you. If you’re experiencing HP Scanner Not Working issue, our trained scanner technicians are smartly known for resolving this issue immediately. As a reliable third party support provider, we use remote technology to identify actual causes of it and apply the right techniques for this issue. Our helpline number is open round the clock to help you for any kind of technical troubles

    ReplyDelete
  37. MyAssignmentHelp being one of the most active assignment help websites has a pool of over 4500+ assignment experts from Australia, UK and US.
    When you opt for the text rewriter tool from MyAssignmenthelp.com, you will have a smooth experience of rewriting the paper. The built-in word rewriter software of our site has been created with advanced technology. As a result, you will always get the most appropriate set of words and expressions to represent your essay topic. So, you should have no doubts about its accuracy.

    ReplyDelete
  38. Keep sharing such a useful information. My Assignment Experts is that they only deal in high-quality plagiarism-free assignments that are the need of the students if they want to score the perfect grades. All assignment help is provided at affordable prices so that the majority of students can use this service to get SQL assignment help. Assignment experts also provide 24X7 customer support to students so that they can come and seek help whenever they need it.

    ReplyDelete
  39. We, at IdealAssignmentHelp, provide exceptional SolidWorks Assignment Help to the scholars residing within the Australia, melbourne,etc we have excelled within the sphere of providing online solidworks assignment help service. we help all those students who need solidworks assignments’. At IdealAssignmenthelp, students will simply convey their requirements and have them fulfilled by highly qualified and skilled subject experts.

    ReplyDelete
  40. I am really impressed through reading this post and urge its writer team to share same level information. Take the association of Assignment Help team as you are in the compulsion to create assignment on committed time to your college professors.
    Assignment Help Online
    Best Assignment Help
    Assignment Helper
    Assignment Help In USA
    Online Assignment Help
    Assignment Help Experts
    Online Assignment Help Services

    ReplyDelete
  41. Hello everyone,
    Hope you all are good! I am a student and looking for online Academic Writing Services but not able to find a trustworthy writer. I have so many assignments to do but have a little time left to submit all my assignments. I can’t do all the assignments by myself. All the assignments are too long and the time- taken. That’s why I am looking for availing of the assignment writing services from the external source. Although, there are so many writers available on the internet. But how to trust them that they would complete my assignments on time, it is getting difficult for me. It may be a chance that they don’t write the plagiarism free assignment.

    ReplyDelete
  42. Assignment Help creates a bridge between a student and assignment writing service provider to lower academic pressure. Connect yourself with greatassignmenthelp to get the best utilization of this bridge by paying a reasonable amount.
    Assignment Help Online
    Online Assignment Help
    Assignment Help Online Services
    Assignment Helper
    Assignment Assistance
    Assignment Help Experts
    Online Assignment Help Services

    ReplyDelete
  43. Amazing post! This blog is very helpful for Nepal Tour Packages. This amazing post gives ideas for Indian travelers how to travel Nepal affordably. I like this post very much. Thanks for sharing this useful blog.
    Nepal Tour Package
    Nepal Tourism Package
    Nepal Tourist Packages

    ReplyDelete
  44. Assignment Help is an effective solution to complete assignment within the due dates even in the US. For more information, login to assignmenthelpshop and discuss your concerns with academic writer.
    Assignment Help Online
    Best Assignment Help
    Assignment Helper
    Assignment Help In USA
    Online Assignment Help
    Assignment Help Experts
    Online Assignment Help Services

    ReplyDelete
  45. Hii everyone,
    Hope you all are doing great!! Actually, I want your help. I am living out of India and want to perfome Festival Pujas of India but don’t know how to perform pujas from here. Does anyone of you know what is the procedure of booking online Occasion Puja ? If you know any reliable website, then please let me know. I would be waiting for your answer. I would be highly grateful for your help.

    ReplyDelete
  46. Thanks for your valuable post. Please post some more related topics. We are on your page only to read your blog because, as you know, there are no other for being here. Like you, we also here for some information of Outlook, so if any of you need any type of help related Outlook then contact our Outlook support phone number or visit the website.
    Is there a phone number for Outlook support?
    How do I contact outlook by phone?
    Outlook customer service number
    Outlook customer service

    ReplyDelete
  47. This write up is really good which gives the clear understanding how to scale up your knowledge stream without any difficulty. If you want to add some valuable impact in your data, you can use the feature of Microsoft office 365 download for getting the best result.Visit here:- https://www.office365download.co/
    office 365 download
    Download Office 365
    Microsoft office 365 download
    download Microsoft office 365

    ReplyDelete
  48. The mentioned points in this guide are very informative and helped me to resolve my issue in a very proficient manner. At the time of solving my query I got stuck at one point, so I called at outlook Support Phone number and the technician who answered my call guided me thoroughly to get rid of my issue. I really appreciate the hardworking of that guy, who helped me during my crisis of time and never leave my side until problem didn’t get resolved. Therefore, I personally suggest everyone to get hold of Microsoft outlook Support if you really want to resolve your query related to your Outlook.
    How do I call Outlook support?
    Is there a phone number for Outlook support?
    How do I contact Hotmail Support?
    Is there a phone number for Hotmail support?
    Change Hotmail Password
    hotmail change password
    change microsoft password
    Is There A Phone Number For Microsoft Windows Support?
    How Do I Contact Microsoft Windows Support?
    How to Contact Microsoft Windows Tech Support?
    How Do I Contact Microsoft Office 365 Support?
    Is There a Phone Number for Microsoft Office 365 Support?
    Is There a Phone Number for Microsoft Office Support?
    How Do I Contact Microsoft Office Support?
    Windows support
    Microsoft windows support

    Office 365 Support
    Microsoft Office 365 Support
    Microsoft 365 Support
    Office 365
    Microsoft Office 365

    ReplyDelete
  49. Are you facing QuickBooks Error h505? If yes, I am available round the clock to help you remotely. I have a higher level of technical experience and expertise of resolving QuickBooks Error H505 completely from the root. I and my QuickBooks experts are available 24 hours to help you in the right ways.

    ReplyDelete
  50. Thanks for your valuable post. Please post some more related topics. We are on your page only to read your blog because, as you know, there are no other for being here. Like you, we also here for some information of Printer, so if any of you need any type of help related Printer then contact our Printer Is Offline Window 10 or visit the website.
    Printer Is Offline
    HP Printer Offline
    Printer Offline Window 10
    HP printer is printing

    ReplyDelete

  51. Class College Education training Beauty teaching university academy lesson  teacher master student  spa manager  skin care learn eyelash extensions tattoo spray

    ReplyDelete
  52. Class College Education training Beauty teaching university academy lesson teacher master student spa manager skin care learn eyelash extensions tattoo spray

    daythammynet
    daythammynet
    daythammynet
    daythammynet
    daythammynet
    daythammynet
    daythammynet
    daythammynet
    daythammynet

    ReplyDelete
  53. I want to thank you for this excellent read!! I certainly loved every little bit of it. I have got you book marked to check out new stuff you post. Our tested and proven writers spend many quality hours researching and writing our customers’ instructions. That’s how they create high-quality custom papers that help our customers better their writing skills. Even the strictest of teachers find it hard to deny those we help the “As” and “A-pluses” they deserve. Find more from My Assignment Help

    ReplyDelete
  54. It’s an amazing piece of writing in support of all the web visitors; they will take advantage from it I am sure. Are you tired of trying out cures for your back pain? Has your back pain brought your life to a stop? Have you tried out several techniques, medicines, sedatives, and therapies, but all of them provide temporary relief? Then Visit Best Inversion Tables for more.

    ReplyDelete
  55. This content is the greatest in internet thank you for sharing. Are you pursuing CIPD and you are experiencing difficulties with getting around the assignment given? If the answer is yes, you need some expert help with your assignments. Choosing our CIPD Assignment Help is a wise decision. We will provide you with the only solution you need for your challenging CIPD assignments in minutes. Visit CIPD Assignment Help for more.

    ReplyDelete
  56. This post was awesome thank you for sharing. GIS (Geographical Information System) are basically systems of computers that are designed to collect, analyze, manage and share geographical and spatial data. Remarkably, almost all environmental applications today utilize GIS. In other words, GIS has gained wide applications in both research, management of wildlife, tracking specific species, informing agricultural practices among a tone of other different uses. Learn more from GIS Assignment Writers

    ReplyDelete
  57. as in most cases the children have problem of having short memories and instructions more than one sentence are likely to confuse the child. write my essay

    ReplyDelete
  58. Now, you can enhance your marks and improve your academic performance using Assignment Help from getassignmenthelponline. If you need expert's help in your assignment or you are facing problem in writing assignments of any kind like Finance Assignment Help Dissertation Help Law Assignment Help Java Assignment Help etc then you must explore https://getassignmenthelponline.com

    ReplyDelete
  59. My Assignment help is the most preferred My Assignment help offers by native writers in Australia. Our proficient writers are available 24x7 to offer best assistance in writing where our services come with a host of offers for returning customers. These services available at an affordable price help you score high and grasp the complex concepts of your preferred academic subject.

    ReplyDelete
  60. ABC Assignment help is the proficient academic writing service holding a resourceful team of more than 3,000 assignment tutors, dedicated to assist students with the authentic academic support. Our team is always available to help students on a plethora of assignments, essays, coursework and more.

    For More - physics assignment help

    ReplyDelete
  61. As of today’s it is very likely that every person have a mobile phone in his hand, and most of the Smartphone is working Android operating system because of its user interface. Sometimes people install some applications on their phones that they might not need but they don't know how to uninstall android App. If you are having the same trouble they don't worry just visit our website and we will help you regarding any of your issues.

    ReplyDelete
  62. Thank you for bringing to a halt my long search topic. I really benefited from your content. If you are experiencing trouble striking a balance between your busy academic life and handling your online classes, you can get help from a professional who will take online classes on your behave. It doesn’t matter the kind of help you need, whether it is passing your exam, finishing your assignments or even the entire classwork, you can get help here at Online Class Help .

    ReplyDelete
  63. The Strippers in the club are naked and some half naked this reflecting immorality in that society. freelance writer for hire

    ReplyDelete
  64. Awesome post I must say and thanks for the information. Thanks again for providing us these unique guidelines it's actually helpful for each person. I appreciate your post and look forward to more. assignment service australia

    ReplyDelete
  65. Nowadays computers are the basic need for many of us. But what good is a computer if it doesn't start when one wants to use it. If you are also having the same trouble with your computer then feel free to visit our website at the HP computer won't boot on windows 10 or call our customer care executive to resolve any of your issues regarding your computer.

    ReplyDelete
  66. If you are searching for the best assignment help service provider than visit https://getassignmenthelponline.com/ this is the bestest and cheapest service provider ever. They provide all subject services according to your requirement.

    please visit
    Math Assignment Help

    Law Assignment Help

    Assignment writing service

    ReplyDelete
  67. Use Assignment Help to connect with Australian academic writers in just a few minutes. Students can take online assignment writing services in Australia when they have no one to ask their queries and find it tough to write your academic papers.
    My Assignment Help
    Assignment Help Online
    Online Assignment Help

    ReplyDelete
  68. This emailing service has many striking features and advanced functions. I am using roadrunner technical support for my business deal hence I depend on this emailing service.

    ReplyDelete
  69. Have you tried the steps given on 123.hp/setup site? Were you not able to follow the process? If not, then it’s not a big deal. There are various customers who fail to understand the technical language. Hence, you can get in touch with our technical professionals. They are having high level of experience and guide their customers in simple language to make them easy to understand. However, to get the correct information you will have to give the correct background details of your HP Printer. The service is available all round the clock, 24*7 to assist their customers.
    123.hp.setup

    ReplyDelete
  70. Schedule tasks and repair jobs for your gadgets with our Geek Squad Appointment. Geek Squad solutions are the best for your applications and appliances.

    ReplyDelete
  71. You can use the Geek Squad Scheduling service and appoint our experts for assistance. Our team is 24/7 ready to help with PC, home appliances and electronics.

    ReplyDelete
  72. Facing issues while making online payments through PayPal don’t get tensed get instant PayPal Customer Support. We are available 24/7.

    ReplyDelete
  73. Thank you for predicting our needs and writing great and informative article. Shade nets are the upcoming synthetic fibers materials that are nowadays preferred in the construction of net houses, greenhouses and preparing seed nurseries. They are mostly used in a broad variety of crops like blackberries, chives, and tomatoes. Visit Shade Nets Purchase and Installation

    ReplyDelete
  74. I totally depend on yahoo mail account. But due to some technical problems, my Yahoo mail account is not working properly. I’m experiencing yahoo mail not working issue. This technical issue is very annoying, so I want to take proper technical care of this technical error.

    ReplyDelete
  75. Freelance Computer Hardware Technician

    With businesses growing larger and more demanding, the need for a global on-demand workforce has never been so urgent.

    ReplyDelete
  76. If you have a wireless HP printer and don’t know how to do wireless HP printer setup then get help from experts. Certified technicians are always available at the support desk to help customers regarding HP printers.

    ReplyDelete
  77. Thanks for sharing this post, We are the most recognized independent and trustworthy third-party technical support company, delivering top quality support and specialized assistance for brother printer users. If you are facing Brother printer in error state, our printer technicians have the best technical skills and extensive experience for solving this error immediately. Our printer technicians are technically known for resolving this technical error within a few minutes.

    ReplyDelete
  78. Get support from professional assignment help in the UK when you don't want to take risk of losing your academic performance. So, connect with experts in the UK and get the best platform to receive outstanding academic writing in just a few steps.

    ReplyDelete
  79. Thanks for sharing this post, I am a technical expert, having the sharp technical capabilities and higher level of experience of solving all types of issues related to Epson troubleshooting issue. If you have any doubts regarding to Epson printer troubleshooting , our techies are technically known for resolving any type of errors related to your device.

    Epson cannot communicate with the scanner

    Epson printer won’t feed paper issue

    Epson Printer Utility Setup Error 1131

    Epson printer drivers for windows 10

    ReplyDelete
  80. If you are looking for updating Garmin GPS unit, you can call online Garmin professionals to provide the best support for solving Garmin updates errors. Our Garmin professionals are skillfully and technically recognized for resolving Garmin updates problems completely from the root. Our Garmin experts are smartly known for updating GPS unit in the simple ways.

    ReplyDelete
  81. If you are facing the issue of Printer driver is unavailable, then there is no need to panic. Follow the below laid steps and het your problem resolved. The option one says that you can update your windows software if updates are available. Second option says that you can re-install your printer drivers once. The drivers must be incompatible with the system otherwise they will misbehave. Apply any of the two options and the problem will get fixed. If it persists, then you can also contact with the printer support expert and professionals.

    ReplyDelete
  82. All HP printers are very reliable and offer great performance during printing and scanning. Sometimes users face HP Printer Error 79 due to outdated printer firmware or corrupted print job. If you have error 79 with your printer then get help from experts to fix it.

    ReplyDelete
  83. Very Nice and Great post. I am glad to see people are still interested of this blog. Thank you for an interesting read more....Private Label Cosmetics Suppliers in India.

    ReplyDelete
  84. online assignment help in Australia that ensures that you meet the deadline every time. It is important that the assignments are written in a way that all the questions asked by your professor are answered and all the objectives are fulfilled. Our subject experts make sure that we do not deviate from the actual context of the assignment and provide the best possible solution without charging an exorbitant amount. Our fair pricing and discounts make our services even more accessible and the best choice when availing the online assignment help in Australia. We consider it a sign of discipline to meet the deadline and submit top quality work within the designated time limit. The subject experts at assignment work help deliver the required output to meet the expectations of our clients every time and make sure that you get good marks for every assignment.
    online assignment writing help
    assignment solution australia
    assignment help australia
    online assignment help
    best assignment help
    assignment work help usa
    assignment work services asutralia

    ReplyDelete
  85. Thank you for your outstanding article. I will be your regular visitor. Are you experiencing difficulty in writing business analytics? Well, you don’t have to experience sleepless nights because of business analytics assignments. At our Business Management Assignment Help Service, we have experienced business analytics writers who can provide you with the much-needed help in writing your assignments. Visit Business Analytics Assignment Help. for more

    ReplyDelete
  86. I recommend to visit my assignment help online website to get our assignment help Sydney service for all your academic tasks. Our website will help you in your studies and if you are thinking to improve your skills, we can help you in that as well. Without thinking much hire our service now!

    ReplyDelete
  87. researchmasteressay is the leading provider of custom essay, term paper & research paper writing service. We provide skilled custom writing services of the highest quality as we ensure that your order is custom written to the first-class standard within your timeframe and at the best price. A skilled writer can handle your essay writing and assignment with the great knowledge of your topic and theme. Custom Essay Writing Services in USA | Best Essay Writing Services in USA | Best Coursework Writing Services | Research Paper Writing Service in USA | Term Paper Writing Services in USA

    ReplyDelete
  88. UK custom essays are the best choice for students looking for professional writers for premium custom writing service. Our custom written essay service only uses top-rated, professional authors. They are all professionals in their field and their reliable services ensure that you get the marks you earn – and get all the recognition you deserve. Just place your order now and make the most of it. custom essay writing service uk | research proposal writing service uk | Best dissertation writing services uk | Best Coursework Writing Service | Best assignment writing service uk

    ReplyDelete
  89. Hello there, great stuff you’ve shared here I will be checking regularly for more articles. But wait! When it becomes hard to manage your online assignment, availing Online Assignment Help is the best choice. Many students often fail to submit their assignment before deadlines and end up scoring low grades because they didn’t have enough time to write the assignment given on time and satisfactorily. Visit Assignment Help . for assistance.

    ReplyDelete
  90. When I try to login into my Hotmail email account, I am failing to login into hotmail email account. I am facing Hotmail email login problems, so I am unable to access my Hotmail account. I don’t have complete technical skills to resolve this login issue. So anyone can suggest the appropriate ways to resolve Hotmail email login error.

    ReplyDelete
  91. How to Troubleshoot Gmail temporary error numeric code 7?
    Gmail error codes are usually temporary in nature; however, some users do find them tough to resolve it. Among many errors related to Gmail, the Gmail temporary error numeric code 7 is the typical one. In case, if you are facing such a numeric error code, no worries, you can get in touch with Gmail executives straight away.

    ReplyDelete
  92. Why Does My Printer Say Offline’ is one of the most asked queries. The printer goes offline owing to multiple reasons. If you have the same question, why does my printer say offline, then you first need to find the root cause of the problem.

    ReplyDelete
  93. Jendral poker adalah agen judi yang terpercaya di server PKV dengan persentase kemenangan yang tinggi.

    Whatsapp: +855 8731 8883

    ReplyDelete
  94. 'I'm highly impressed by the piece of thoughts you have shared on this portal. all the best
    connect us on Assignment Help can shed your burden of assignments with a return of qualitative assignments.
    Online Assignment Help
    Programming Assignment Help
    Management Assignment Help
    assignment experts
    Networking Assignment Help

    ReplyDelete
  95. Mkvpoker Adalah Situs Judi Poker Terbaik Dengan Taruhan Uang Asli Yang Menjaminkan Kemenangan Yang Tinggi
    Hub WA : +855 9637 31855

    ReplyDelete
  96. Mkvpoker Adalah Situs Judi Poker Terbaik Dengan Taruhan Uang Asli Yang Menjaminkan Kemenangan Yang Tinggi
    Hub WA : +855 9637 31855

    ReplyDelete
  97. This post is really good which inspires you how to cultivate your knowledge stream. For pouring the relative style in your upcoming post, you can use Download Office 365 link to execute the file without any hindrance. Visit here:- https://www.office365download.co/
    office 365 download
    Download Office 365
    Microsoft office 365 download
    download Microsoft office 365

    ReplyDelete
  98. Avail best Assignment help services without going out of your budget. Assignment writing services are beneficial for de-stressing and reducing students’ workload amazingly. If you have stress of writing your academic papers, transfer you work to me and pay attention to your studies.
    Read More:-
    Online assignment help

    ReplyDelete
  99. Looking for the latest Bollywood celebrity news? It's not easy to find but there are tons of ways to find this. The first thing you need to do is to search the Internet for the celebrities that are in your area, people who are related to your country, and individuals in that area are the best ones to search since they have the most up to date information. Then you can put the name in a search and get the results from Google or Yahoo. Once you find the name you like you can get the most updated news articles from websites where you can also get the latest Bollywood gossip today. In general the most accurate way to keep up with Bollywood news is through social networking sites. Some websites allow you to upload the news article you want, and others allow you to add comments on them. This is the easiest way for finding the latest Bollywood celebrity news and updates. You can also find the celebrity news and latest bollywood gossip today on your favorite site at Bankofbollywood. latest bollywood celebrity news | latest bollywood gossip today | todays bollywood gossips | latest entertainment news today | today's entertainment news headlines in english | political breaking news | latest business news headlines

    ReplyDelete
  100. If you are unable to find the solution to resolve Yahoo mail problems not receiving emails then you can contact the experts to get the solution. When none of the options helped you as mentioned above, then taking help from Yahoo technicians can resolve the issue in just a few minutes.

    ReplyDelete
  101. Are you in need of chemistry assignment help service? Then get it from ABC Assignment Help, as they have a pool of specialist’s team from various universities in almost every field. They are enough proficient in writing any kind of assignment according to the requirements of students.

    ReplyDelete

  102. Promo 20rb dibulan penuh berkah yuk buruan hanya di AGEN POKER ONLINE new member diberi 20k skuyy wa :+85510903838

    ReplyDelete
  103. Mkvpoker Adalah Situs Judi Poker Terbaik Dengan Taruhan Uang Asli Yang Menjaminkan Kemenangan Yang Tinggi
    Hub WA : +855 9637 31855
    Kunjungi Situs Kami Di : Mkvpoker.pw

    ReplyDelete
  104. Mkvpoker Adalah Situs Judi Poker Terbaik Dengan Taruhan Uang Asli Yang Menjaminkan Kemenangan Yang Tinggi
    Hub WA : +855 9637 31855

    ReplyDelete
  105. What a wonderful piece of article I have bookmarked for future checkup. A capstone paper is normally a lengthy academic document that demonstrates a student’s ability to conduct independent research and to communicate the findings or the results in an academic manner. This type of academic paper follows a conventional format and it is normally written by students during their final year of pursuing a given programme. Learn more on Capstone Project Writers .

    ReplyDelete
  106. Use Java Programming Help if you have less knowledge of coding and programming language. If you find tough to compose your Java papers before the due dates, get expert's help using the option of online Java programming services.
    Java Assignment Help

    ReplyDelete
  107. Secret word is the way in to your messages and on the off chance that they get taken, lost or traded off it be exceptionally extreme for you to recover it. It is constantly proposed to change your secret phrase as often as possible to limit the hazard. In the event that you are an AOL email client and you wish to change aol password word on your apple gadgets at that point basically follow the underneath referenced advances and complete it without anyone else.

    ReplyDelete
  108. Moxie Pest Control is one of the best company, and one of the reason is the safe methods of treatment that they use. To try, call them now.

    ReplyDelete
  109. To get the correct of How many pages is 1000 words, keep the essential pointers in your mind. For experts’ answer on this question, use assignment help services and connect with professionals.
    1000 words to pages
    1000 words is how many pages

    ReplyDelete
  110. https://m-phi.blogspot.com/2011/10/inconsistency-of-pa-and-consensus-in.html?showComment=1590827871080#c8706309486740133986

    ReplyDelete
  111. According to a Pest’s related survey, over 67% of American homeowners experience pest problems and about 22% of them face some structural damage caused by pests. The best way to stop the continuous attack of these little creatures is professional pest control treatment. However, only a good and certified company can provide you best and long-lasting results, such as Clarks Pest Control CA. Clarks Pest Control CA offers a range of offers a variety of services and solutions to its customers to fight with pests situations. The company serves residents and businesses.

    ReplyDelete
  112. Assignment Help Hong Kong is the best option to complete academic papers without any stress. If you don’t want to degrade your academic performance, make sure to connect with the professional service provider of assignment writing help.
    Assignment Help Online
    Online Assignment Help

    ReplyDelete
  113. Promo 20rb GRATIS yuk buruan hanya di AGEN POKER ONLINE new member diberi 20k skuyy wa :+85510903838

    ReplyDelete
  114. It’s truly a nice and helpful piece of info. I am satisfied that you just shared this helpful info with us. Please stay us up to date like this. Thank you for sharing.

    ReplyDelete
  115. Microsoft Office installation is essential tool for any PC inside this generation. It provides quite useful tools such as MS-Word, Excel and PowerPoint that are crucial for creating your task quite easy associated with company. This attribute makes Office.com/setup one of the greatest software bundle.
    office.com/setup
    office.com/setup
    office.com/setup
    mcafee.com/activate

    ReplyDelete
  116. Very Interesting! Some very valid points! I appreciate you writing this post and also the rest of the blog is really good.

    ReplyDelete
  117. Hoardings in Delhi NCR, Outdoor Advertising Agency in Delhi NCR, Outdoor Hoarding Advertising Service in Delhi NCR is the best way to promote your business as those who handle to get the target audience's maximum exposure. Outdoor billboard advertising in Delhi NCR definitely has an advantage over other means as it is a cost-effective way of capturing the attention of a potential customer to the products or services.
    We are popular in providing a broad Outdoor Hoarding collection which is used for advertising purposes in Delhi NCR. These stands are assembled according to specified industry standards using high-grade raw material and cutting-edge technology. These stands are available in various technical configurations and can also be customized to suit the varied needs of the customers.

    ReplyDelete
  118. What a great piece of content giving a good insight on topic at hand, Need more support for your lower back? Interested in something cushy to put your feet on? Comfortable pillows made for lumbar support are perfect for these situations. Besides comfort, a pillow allows you to express your style. Read more on Throw Pillow Covers

    ReplyDelete
  119. Your appearance is a reflection of your personality try these cool boho dresses and accessories @ bohemian land to shine
    topic

    ReplyDelete
  120. give a brand new look to your car by customized car accessories and toys @ auto x tools
    pillows

    ReplyDelete
  121. Myassignmenthelp offers customized and unique assignment help through actual professionals. Our team of PhD. certified experts strive to meet students’ expectations along with University guidelines to ensure high scoring assignments every time. Along with adept writing help we offer professional proofreading, editing, free samples and exam help to assist students in achieving their career aspirations.

    ReplyDelete
  122. Myassignmenthelp offers customized and unique assignment help through actual professionals. Our team of PhD. certified experts strive to meet students’ expectations along with University guidelines to ensure high scoring assignments every time. Along with adept writing help we offer professional proofreading, editing, free samples and exam help to assist students in achieving their career aspirations.

    ReplyDelete
  123. Nice information. Thanks for sharing content and such nice information for me. I hope you will share some more content about. Please keep sharing!

    ReplyDelete
  124. Thanks for sharing it is important for me. I also searched for that from here. Visit our site Trend Micro nummer belgie

    ReplyDelete
  125. ELTS band score calculator

    converts total number of correct answers into approximate IELTS band score. The correct answers ranging from 0 to 40 and band scores from 0 to 9. It simply gives you an idea about your band score in the IELTS exam by taking practice tests at Gratis Academy. The approximate or raw test scores required to get a particular band score in Listening and Reading tests vary slightly in General and Academic exam modules. Each module is rated by Cambridge for a particular level of difficulty which determines the requirements of your raw scores.
    ielts band calculator

    ielts calculator

    ielts score calculator

    calculate ielts score
    band score calculator
    reading band calculator
    ielts score calculation

    ReplyDelete
  126. You can rely on the experienced and the dedicated RoadRunner emai ltech support for creating your account or troubleshooting problem related to your email account. You can call a 24/7 available team anytime or drop an email with your problem mentioned in the subject line.

    ReplyDelete
  127. Assignment Help services in Ireland will help to meet you with professional academic writers without roaming around the city. Enhance your skills and writing attributes using experts’ assistance by placing your order online in Ireland. 

    ReplyDelete
  128. While using AOL sometimes people used to error messages while AOL login, if you are having the same problem then you can visit us to know more about how to eradicate Aol error message while using AOL services.

    ReplyDelete
  129. To use outlook on your iPhone one must install the office 365 setup on their iPhones to use outlook. To know more about this feel free to visit us at How to setup office 365 email on iPhone.

    ReplyDelete
  130. While working on a computer sometimes one has the need to get the screenshot of the current window to get the screenshot one must know about How to screenshot on HP laptop. If you also want to know about how to take a screenshot then you must follow the link mentioned here.

    ReplyDelete
  131. Sometimes while using a Laptop the touch screen of the laptop might stop working. If you are also having the same issue with your laptop why wait just visit fix hp laptop touch screen not working.

    ReplyDelete
  132. Cash App provides a better and simple way for transfer money online. Sometime new user stuck Cash App Online Login. If you are also facing this issue. Please connect with Cashapp Login for more information. Our team of technician and expert help you to How to Sign into Cash App.
    Cash App Login

    Cashapp Login

    Cash App Login Online

    Cash App Sign In

    Cash App Login Website

    Cash App Online Login

    Login into Cash App

    Login Cash App

    How to Sign into Cash App

    How to Login to Cash App


    ReplyDelete
  133. Great post!!!! I will defiantly use this information in my future and share with my friends and colleagues. Thanks for sharing this post. If you have any problem related with Cash App transfer failed then visit these links.
    Cash App Transfer Failed
    Cash App Payment Failed
    Cash App Failed for My Protection
    Cash App Canceled for your Protection
    Cash App this Transfer Failed
    Cashapp Payment Failed
    Transfer Failed Cash App
    Cash App Payment Failed for My Protection

    Thank You

    ReplyDelete
  134. I admire your art of writing. Physical therapy is a specialty in the field of health care, which utilizes exercises, massage, heat treatment, and such physical methods to treat an injury, deformity, or a disease instead of using surgery or drugs. If you are interested in studying such interventions, then you should pursue physical therapy studies in an institution of higher learning. Read more on Physical Therapy Capstone Writers

    ReplyDelete
  135. Thank you so much for sharing these amazing tips. I must say you are an incredible writer, I love the way that you describe the things. Please keep sharing

    How to Change AOL Password on iPhone
    Change AOL Password on iPhone

    ReplyDelete
  136. Very Nice and Great Blog post!! Blaze Hotels best quality and service provider in India. read more information....Hotel management services

    ReplyDelete