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Hi, this is Jose Ferreiros. I find that the introduction to the whole article reflects very directly a new interpretation of the emergence of logicism that I began to put forward in the 1990s. Yet there is no reference to my papers on the topic. So I would greatly appreciate it if you took care to add references. Specifically, Dedekind's role in the early history of logicism was quite absent from older articles before 2000; insistence on the crucial role of the derivation of the real numbers in Dedekind's path to logicism is also a characteristic of my work. You may find that already in a 1996 paper published in Arch. for Hist. of Exact Sciences ('Traditional logic and the early history of sets') and then in the book Labyrinth of Thought. — Preceding unsigned comment added by 90.162.182.80 ( talk) 15:44, 3 March 2015 (UTC)
By the way, let me support the next comment when it says that the introduction is somewhat unclear. Logicism is about the reduction of math to logic, not about math being an expansion of logic. Also, the fact that Gödel's theorem is proved "by logic" is quite irrelevant to its significance for logicism. One may say that the incompleteness of formal systems of mathematics is not an insurmountable obstacle for logicism, but it remains true that all the early logicists (Dedekind, Frege, Russell) expected the basic foundational systems to be complete. And if the logical system is incomplete, the sense in which logic is "formal" becomes a question. [Let me also thank you guys for the work you do for Wikipedia.] — Preceding unsigned comment added by 90.162.182.80 ( talk) 16:16, 3 March 2015 (UTC)
Hi. I've just read through this article, and it's a real mess. A huge amount needs to be done to it, possibly up to the point of a complete rewrite. Before I criticise it, I should mention that I have just completed a PhD thesis whose topic was a particular strand of neo-logicism. I have also taught some philosophy of mathematics at university level. (I don't mean to boast or wave credentials about. I just want to show that I'm familiar with the material.) I think that this qualifies me to criticise the article.
Anyway, after a read through, here's what I think are some problems with the article:
...
I could go on, but I'm getting tired. It is my opinion that the article could do with a complete rewrite; it is largely very rambling, for the most-part unreferenced, and the notation is completely non-standard. I'm adding a template to this effect now, partly so that people will see my comment here, and hopefully chip in their 2 cents.
I'm happy to have a go at a rewrite. I believe that I am qualified to do so. I'm going to start a version of the article in my sandbox. Then, at the point where I think I have something better than this article (but obviously with ample scope for revisions), I will suggest replacing this article. Would people be opposed to such an idea?
I would suggest a structure for the article as something like the following:
I am happy to have a go at this, and happy to ask for help from others. Again, would anybody be strongly opposed to this? Jdapayne ( talk) 20:35, 7 February 2013 (UTC)
Keep in mind that the article's title is "Logicism", not "Neologicism". So you should be careful to keep your perspective tuned to logicism, especially with regards to the historical; that's why we came to this article, to find out what we could about historical logicism; if we wanted to read about neologicism we'd be reading about it, somewhere or other. I agree with you that neologicism should be split off from this article. Secondly, be sure you have sources for everything. The third comment is a caution: you are apparently new to wikipedia. We've all been newbies, once, and we've all made the same mistakes -- over-enthusiastic, overly critical. My advice to you is to go slow, don't tear everything you see to shreds. At least at the beginning of your efforts, try to find some good in the article and repair the blunders, i.e. work with what's there, build on it . . . slow and steady wins the race. Lastly, keep in mind that a "rewrite" such as the one you are proposing will, someday, be applied to your work, work you will not get an ounce of credit for -- you do not own even the tiniest mote of the work you put into this, and it will get the exactly the same respect you give the work you find here now. Bill Wvbailey ( talk) 02:06, 8 February 2013 (UTC)
You sound like you will be a good editor. Are you familiar with "sandboxing"? I've created dozens of them whenever I want to really have at something, but don't want to clutter the talk page e.g. with quotes, sources, etc. They're fake pages under your "namespace" e.g. the one I used one on this article: User:Wvbailey/Logicism, which might be useful, actually. You would create a new article e.g. User:Jdapayne/Logicism and work it to your heart's content with no need to write "edit summaries" etc.
Here's some background (and some rebuttal) about the article as it is now, in no particular order:
Hope this helps, Bill Wvbailey ( talk) 20:51, 8 February 2013 (UTC)
That does help. Thanks. It was rash of me to suggest a rewrite straight away. What I will do instead is mainly to try and fill in stuff about Frege's logicism - both the philosophical part and the construction - since I think that is something that the article is mainly lacking at the moment, and also the part that I know most about. This might require a bit of reorganising as well to split off the Frege from the Russell bits. I think the idea to separate the construction of the natural numbers out seperately is a good one. I won't try to do that yet, but I'll try to make is so that it's easy to do if needed (by keeping the various bits separate in this article). I've started sandboxes at User:Jpayne/Logicism and User:Jpayne/Logicist constructions of arithmetic for this purpose.
If you're still working on this article, an additional sources that you might want to look at for are the Oxford Handbook of Philosophy of Mathematics and Logic, which has an article on the logicism of Dedekind, Frege. There are also a few articles in the Stanford Encyclopedia of Philosophy which may be useful - especially one on Dedekind, at http://plato.stanford.edu/entries/dedekind-foundations/ , which might help for the Dedekind stuff. I'll have a look at these - and follow up other references - after I've witten some stuff on Frege. Jdapayne ( talk) 10:13, 11 February 2013 (UTC)
Can someone explain what it means for mathematics to be "reducible to logic" (opening sentence)? I'm a mathematician and I have no idea what this means. 158.109.1.23 ( talk) 14:18, 21 February 2008 (UTC)
Is Neo-Logicism meant to be a different article? Because otherwise it references to itself 8/. Fephisto 22:31, 8 July 2006 (UTC)
I am thinking of annihilating this page and rewriting it. Would it be a good idea to move most of the current (not very NPOV) content to a new place, like 'Logicism and Godel's theorem' perhaps? (I'm new here.) -- Toby Woodwark 20:37, 2004 Mar 18 (UTC)
Absolutely not. B 00:15, Apr 26, 2004 (UTC)
I find this page very inaccurate and subjective. If an expert in the area is considering rewriting it I would definitely agree with that.
All right, I'm going in. I don't know if this'll come out perfectly, but I do agree that the second half of the article is extremely subjective, if not downright wrong. I'll try to clean it up now, then...
I removed: "Modern philosophers believed that proof of this theory was the means of banishing the befuddlement...". The idea expressed is probably inaccurate for many of the people usually associated with Logicism.
I also removed: "with sucess except for the paradox of trying to formulate a logical definition of natural numbers in terms of classes". Though it certainly is suspect to use a notion of class to give a logical definition of number, I don't see the sense in which this is a paradox. Wjwma 18:48, 2 August 2005 (UTC)
I modified the sentence alleging that Godel's Incompleteness results undermine logicism. Given certain assumptions and certain formulations of logicism, it is true that the incompletness results undermine logicism, but these positions are controversial. Wjwma 18:21, 3 August 2005 (UTC)
This page is badly in need of some references. Otherwise, it is subject to being deleted per WP:NOR. -- noosphere 09:27, 14 April 2006 (UTC)
I changed the word 'valid' to the word 'alive' in the paragraph on Goedel's Incompleteness Theorems because [especially in a mathematics and logic context] valid means something like 'necessarily truth preserving'. To say that logicism necessarily preserves truth despite being a contested position seems too strong a claim to make in an encyclopedia article. Feel free to change the word to something more appropriate than 'alive' if you can think of something. Taekwandean ( talk) 12:41, 6 January 2011 (UTC)
Just to clarify, I am not saying that the Incompleteness Theorems are not valid, but rather that the philosophical position Logicism may not be. There is quite a lot of literature on that topic, but you might, for starters, take a look at some essays by Thomas Ricketts ( http://www.pitt.edu/~philosop/people/ricketts.html) or Peter Clark ( http://www.st-andrews.ac.uk/philosophy/dept/staffprofiles/?staffid=98) to get a sense of some direct objections. More indirectly, one might think that if the foundations of mathematics are some fundamental intuitions of time and self (something like a Brouwerian kind of intuitionism), then it would certainly not be the case that logicism was necessarily truth preserving (since in this case maths would be reduced to intuitions, not to logic).
But, I think we are likely in agreement, just talking past each other. I am not trying to say that logicism is wrong. I am not trying to say that the incompleteness theorem are invalid. I am, however, trying to draw a distinction between necessarily truth preserving on the one hand, and an active position worth of consideration on the other. So, I still think that the word valid should be changed, but perhaps others could weigh in on this issue? Taekwandean ( talk) 19:45, 7 January 2011 (UTC)
Good, we agree that Goedel does not change the status of the logicism claim. I take it that this is no longer the dispute. What I am trying to change is the wording of the sentence that communicates just this fact that we are not in disagreement about. The word I object to is 'valid'. In most logic and math contexts this word means 'necessarily truth preserving'. It is hard to see how a philosophical position (logicism) that is disputed (by, e.g. the people that I cited among others) could be necessarily truth preserving. This does not mean that logicists are wrong. It just means that in an encyclopedia article we might not want to use such a strong word. But, if you feel strongly that it is necessarily truth preserving, I'll leave it alone. Taekwandean ( talk) 11:37, 8 January 2011 (UTC)
(Unindenting)First, widely held by who? By people who work in foundations or by people who do math in some form? The former are the ones that are important in this context. Second, we are not talking about the math most people do when we are talking about reduced to ZFC, but literally everything that can be termed mathematics; so that's a huge claim. Third, what about large cardinal axioms? If all of mathematics can be reduced to ZFC, then are lca's all undecidable and that's it? That would be a minority opinion. Now, suppose that the Riemann Hypothesis is undecidable using ZFC, but that there is some seemingly unrelated purely set theoretic axiom K so it is true in ZFC + K. If everyone adopted K, would we then start saying that all of math is reducible to ZFCK? What makes ZFC the foundation for all mathematics apart from common convention? My point: ZFC is not implicitly The Foundation, it is a foundation. But this is not the topic at hand, what Frege did with higher order logic and arithmetic reminds me a lot of how arithmetic is done in the Lambda Calculus, the sentence I removed did not seem to be talking about this kind of work. It sounded like it was saying Logicism is valid since things are proved using logic; most of the conversation above seems to be saying this too, this is not Logicism, which is my whole complaint here. If this is not what you mean, then you need to write better. Finally, though it is not directly relevant to the article, you never answered my question about what 'Logic' you are talking about. Do you mean that all of math reduces to first order logic, second order logic, higher order logics all together, some modal logic, etc? Further, this page doesn't really address what a logic is, or what logic in general is; which would be fine, if you weren't talking about all of mathematics (in the philosophical total sense of the term) reducing down to it. I'm not trying to be an ass, but your responses are really missing the point of my objections, I did not come here to attack Logicism, I came here and saw problems with the article and, seemingly, with the people editing it. 71.195.84.120 ( talk) 16:06, 31 July 2011 (UTC)
Intent of these entries: The intent here is to figure out exactly what “Logicism” is and how it came about. I'm only discussing "logicism" here, as it is used in the literature. I am not considering "neologicism".
This will be a work in progress. This is going to take a long time to research. See more at User:Wvbailey/Logicism. .
Philosophy of mathematics: In general what we're dealing with here is Russellian epistomology (see below) with application to the philosophy of mathematics. So we need to look at both the "philosophy of mathematics" and "mathematical philosophy", not at all the same thing. Plus there's Russellian monism; see below.
Mathematics applied to philosophy: If "logic" is to be the core of "mathematics" and "mathematics" is to be the core of philosphy, then "logic" is the core of "philosophy". Thus we have Analytic philosophy. The problem RE philosophy appears to be the epistomological aspect.
The progenitors of logicism: The entry re Analytic philosophy includes Frege, Whitehead and Russell as its progenitors; G-G also includes Schroeder, Dedekind, and Pierce to lesser degrees (G-G 2000:249, see quote under Schroeder below). Plus we have to include the influence of Peano on Russell (G-G 2000:249) by his own admission in his 1901.
Epistemology: see entries below re conflation of Russellian epistemics and mathematics.
Monism, neutral monism: Russell was a neutral monist. See User:Wvbailey/Logicism.
Sources: I'm copying a bit of this from Trovatore's talk page. But to see more, reference in particular Grattan-Guinness 2000:411-506, i.e. Chapter 8 The Influence and Place of Logicism, 1910-1930 and Chapter 9: Postludes: Mathematical Logic and Logicism in the 1930s (pages 507-555), Chapter 10: The Fate of the Search(page 556ff) with various subsections titled 10.1.2 "The timing and origins of Russell's logicism", 10.2 "The Content and Impact of Logicism". His history basically ends with 10.2.3 The fate of logicism.
Most of this will be derived from the huge history in Grattan-Guinness 2000 but I’ve included other sources too. Grattain-Guinness 2000:556 begins his last substantive chapter 10 with this dreary conclusion: "The chapters ends [sic] with a flow-chart for the whole story and some notes on formalism and intuitionism [footnote 1, see below], before locating symbolic logic in mathematics and philosophy in general, and emphasizing the continuing lack of a definitive philosophy of mathematics " [my boldface].
--
Role of Peano, Peano's influence on Russell: TBD [see G-G 2000:250: "It [Schroeder's 1897 paper, his logicism] constrasted greatly with the organic construction of logicism from mathematical logic, already tried by Frege and soon to be adopted by Russell under influence from Peano (§6.5)."
In particular G-G observes that the criticism of Schroeder's 1897 included some important points that "would be of major importance for Russell", and also some changes in notation i.e. substituting iota 'ι' for "the", ⊃ for Peano's Ɔ, 'Cls.' for Peano's 'K'. (G-G 2000:251).
Fregian logicism: TBD
Schroeder's logicism: G-G states here that Schroeder, like Hilbert, wanted a methodology to transform all of mathematics to a routine. Of course this is what Goedel proved could not be done. I've bold-faced this part of the quote:
From Kleene 1952:
From Eves 1990 (notice that he seems to have borrowed from Kleene !):
Livio 2011: In the latest Scientific American article there's an article by Mario Livio "Why Math Works" wherein he discusses two -isms only: Formalism and Platonism (August 2011:81) and tries to answer the question about whether mathematics is intrinsic to the universe and discovered by mankind (Platonism), or whether it is Formalistic in nature -- i.e. invented by mankind. He concludes both seem to be the case.
Livio is a bit perplexed by this universe of ours: "Why are there universal laws of nature at all? Or equivalently: Why is our universe governed by certain symmetries and by locality? I truly do not know the answers . . ." (p. 83).
Unlike Livio, Grattain-Guinness recognizes three mathematical philosophies (footnote 1, page 556: "It is strange that the names for the three main philosophical schools were already in use in ethics (Clauberg and Buislav 1922a, 161). Ethical grounds for exercising the will were 'logicistic' if their consistency was held to be morally sufficient; ethical norms were 'intuitionistic' if they were held to be inborn rather than acquired, and 'formal' if they came through general principles rather than individual objects."
Platonism: As to where Platonism falls in this scheme of things, I have no idea. Perhaps under "logicistic"?
Mancosu on Brouwer's Intuitionism: I found a great quote that corroborates my personal opinion that Logicism is a "practice" rather than a philosophy. Mancosu derives this from Brouwer's 1907 The Foundations. As quoted from Mancosu 1998:9 --
Wittgenstein versus Russell: From G-G's "8.2.5 Russell's initial problems with epistimology, 1911-1912 --
Russell's 1914/1926/1929 "Our knowledge of the external world . . ." :
American criticism of Russell 1914 "Our knowledge of the external world . . .":
Bill Wvbailey ( talk) 00:25, 3 August 2011 (UTC)
Annotated Bibliography (Sources):
In the first section, the article says ' theorems of mathematical logic [...] can be proven using the fundamental theorem of arithmetic (see Gödel numbering) '. A reader would be forgiven for taking that to mean that the crux (cruces?) of the proofs of many theorems of mathematical logic is an appeal to the Fundamental Theorem of Arithmetic. Which is of course not the case.
I suspect what the author had in mind is how, in its setup phase, Gödel's proof of his First Incompleteness Theorem uses a simple construction with primes and exponents to in principle assign a natural number to any well-formed formula (wff) in the formal language. The Fundamental Theorem of Arithmetic guarantees the uniqueness of this assignment: if a given natural number corresponds to any wff at all, then it corresponds to exactly one of them.
The result of this construction is of course used in the deep part of the proof, when one feeds the Gödel Sentence its own assigned number through its free variable, but the proof in no way follows directly from the Fundamental Theorem of Arithmetic.
Would the author please reword his statement in order to prevent such unwarranted interpretation of it. The fact that each whole number has a unique prime factorization is in no way a silver bullet to help us prove theorems of mathematical logic. — Preceding unsigned comment added by Schumacher peter ( talk • contribs) 17:46, 13 September 2016 (UTC)
---
This article makes some pretty strong claims for an encyclopedia. First of all, that the incompleteness theorem undermines logicism because it shows that there are undecidable statements is a very subjective position and I think this should be emphasized. The fact that some propositions can neither shown to be true nor false within a particular deductive system does not impede the reduction (or rather translation) of informal mathematics into formal deductions.
Second, the sentence "Therefore, any claim that logicism remains a valid concept must strictly rely on the dubious notion that a system of proof based on man-made models is precisely as powerful and authoritative as one based on the existence and properties of the natural numbers." does not make any sense. I have yet to see any mathematics that is not man-made (proof assistants are man made as well of course). The sentence also seems to claim that the existence and properties of natural numbers are not man made, which might be true from a Platonist point of view, but I doubt that this would be the position of adherers of logicism. This sentence seems like an ill-founded, biased statement about logicism.
The strong position against logicism taken by this article in the introduction of an article does not seem to follow the principles outlined by wikipedias neutral point of view. /info/en/?search=Wikipedia:Neutral_point_of_view — Preceding unsigned comment added by 2601:580:8205:183F:ED7E:783:7B77:B04A ( talk) 21:57, 8 March 2017 (UTC)
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==missing in Wiktionary==
=logicalism=
<code>Wiktionary code</code>
==English==
===Etymology===
{{suffix|en|logical|ism}}
===Noun===
{{en-noun|~}}
# {{lb|en|philosophy}} The doctrine that any possible real world has logical foundations.
====Synonyms====
* {{sense|philosophy}} {{l|en|metaphysical logicism}}
* {{sense|philosophy}} {{l|en|materialism}}
* {{sense|philosophy}} {{l|en|philosophical materialism}}
* {{sense|philosophy}} {{l|en|physicalism}}
* {{sense|philosophy}} {{l|en|metaphysical naturalism}}
====Antonyms====
* {{sense|philosophy}} {{l|en|idealism}}
====Hypernyms====
* {{sense|philosophy}} {{l|en|monism}}
* {{sense|philosophy}} {{l|en|realism}}
====Translations====
{{trans-top|philosophical position}}
* Finnish: {{t|fi|logikalismi}}, {{t|fi|metafyysinen logismi}}
* Galician: {{t|gl|logicalismo|m}}
* Greek: {{t|el|μεταφυσικός λογικισμός|m}}
* Italian: {{t|it|logicismo metafisico|m}}, {{t|it|logicalismo|m}}
{{trans-bottom}}
====See also====
* {{sense|philosophy}} {{l|en|dualism}}
====See also====
* {{l|en|idealism}}
The fact that there's no [omni]universal axiomatics = no omniaxiomatics = there isn't the axiomatic system which includes all the infinite possible (even the infinite weird experimental ones we can make) axiomatic systems, [that fact] is very important and fundamentally related both in logicism (mathematical logicism) and in logicalism (metaphysical logicism).
2A02:2149:8B63:D400:BD1F:56E8:B586:E393 ( talk) 03:49, 9 November 2023 (UTC)
Some neologicists claim that the axiomatic system of all axiomatic systems (omniaxiomaticity/omniaxiomatics/universal axiomatic system) doesn't exist because it would include mutually exclusive axiomatic systems; and the set of all sets doesn't exist (thus it cannot be a set).
Some neologicists/ modern logicists and other mathematicians work on experimental axiomatic systems. Infinite axiomatic systems are logically possible. The vast majority of logically possible axiomatic systems aren't particularly useful. Mathematics is a proof system (see: John Stillwell). Allomathematics is mathematics based on different axiomatics. A true allomathematics must have some mathematical use, for example it can be better than common mathematics for particular types of proofs, computer chip building, etc. Physics isn't a proof system but a substantiality system, and the axiomatic prerequisites for physical foundations/ of the physical foundations (in US English) include a more linked axiomaticity = axiomatic foundations (program-based axiomatics and not a list in order the physical foundations are a specific entity kernel and not diffused axioms logically unlinked that they could be otherwise thus cannot make solid physical foundations). Physical axiomatics requires incorporated the forms of entropy (thermodynamic entropy and informational entropy) otherwise we have timeless stationary geometry and not spacetime. Physical axiomatics isn't based on crystal clear axioms but on engaged axioms with other axioms and with necessary endosystemic interpretations. Quantum mechanics meets the prerequisites for physical foundations, thus the quantum foundations is quantum mechanics. It's supposed vagueness is mechanisms of engagement of other procedures and interpretations because actuality is always relational. The self-causation/ self-causality criterion is met by a logical axiomatic program kernel which is self-caused due to being logical and the quantum foundations isn't tautological to the Big Bang. The Big Bang is a very important event but it's not the quantum foundations. In cyclic cosmology infinite big bangs exist; the universe expands till a big rip so mass–energy producing which collapses into black hole center substance = maximal degeneracy chromodynamic superfluid which fragments into inflation particles and normal big bang (homogenous and isotropic) begins (immediate big rips don't produce homogenous and isotropic big bangs and violate the axiomatic prerequisites for physical foundations). (Versions of cyclic cosmology are mere guesses but they are important to understand the difference between big bang = important event and the logical foundations of substantiality). Not one foundations of logic exist. The ideal Turing machine should have the infinite transcribers for different axiomatic systems. Some axiomatic systems are wholly transcribable, and others in a case-by-case manner. Mathematics can describe all axiomatic systems but many axiomatic systems aren't flexible enough to be able to describe others. Thus many axiomatic systems aren't transcribable to others. The idealized Turing machine isn't supposed to merely calculate simple things but it should be also able to solve difficult mathematical problems which require non-self-evident techniques (otherwise we don't have a true Turing machine because it fails in some categories of calculations). The Turing machine is impossible to be locally realised because the required mathematical techniques are infinite, and infinity is never realisable because in the axiomatic prerequisites for physical foundations infinity isn't relational to specific procedures in a manner which leads to substantiality = existence. Existence cannot ever be infinite. Existence can be linked to infinite phenomena but it cannot be infinite in itself. — Preceding unsigned comment added by 2A02:2149:8B83:6500:7813:142C:413A:4290 ( talk) 09:15, 11 March 2024 (UTC)
In modern forms of logicism infinite different logical systems are possible. Not all logical systems have to be proof systems like mathematics (see: John Stillwell on proof).
Comparative logicism compares the traditional logicism, the forms of neologicism including the very weird experimental non-proof logicisms (infinite are possible).
Comparative logicism is important because "some ways to build parts of logical systems are more effective than others". Thus we can say that traditional logicism is wrong for being very idealistic, but comparing elements of various forms of logicism remains important because at least partially some logical foundations seem more effective than weird experimental logical foundations. It's something like the Feynman diagrams. We can create very needlessly weird logical foundations, but some more traditional logical foundations are more probable. But a physical foundations of any universe would include some more probable and other less probable foundational parts (comparative logicism understands that all forms of logicism can be part of the axiomatic foundations of universes = physical laws). — Preceding unsigned comment added by 2A02:2149:8BE9:C00:F0A3:E95F:3E9E:9FBF ( talk) 17:57, 13 April 2024 (UTC)
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Hi, this is Jose Ferreiros. I find that the introduction to the whole article reflects very directly a new interpretation of the emergence of logicism that I began to put forward in the 1990s. Yet there is no reference to my papers on the topic. So I would greatly appreciate it if you took care to add references. Specifically, Dedekind's role in the early history of logicism was quite absent from older articles before 2000; insistence on the crucial role of the derivation of the real numbers in Dedekind's path to logicism is also a characteristic of my work. You may find that already in a 1996 paper published in Arch. for Hist. of Exact Sciences ('Traditional logic and the early history of sets') and then in the book Labyrinth of Thought. — Preceding unsigned comment added by 90.162.182.80 ( talk) 15:44, 3 March 2015 (UTC)
By the way, let me support the next comment when it says that the introduction is somewhat unclear. Logicism is about the reduction of math to logic, not about math being an expansion of logic. Also, the fact that Gödel's theorem is proved "by logic" is quite irrelevant to its significance for logicism. One may say that the incompleteness of formal systems of mathematics is not an insurmountable obstacle for logicism, but it remains true that all the early logicists (Dedekind, Frege, Russell) expected the basic foundational systems to be complete. And if the logical system is incomplete, the sense in which logic is "formal" becomes a question. [Let me also thank you guys for the work you do for Wikipedia.] — Preceding unsigned comment added by 90.162.182.80 ( talk) 16:16, 3 March 2015 (UTC)
Hi. I've just read through this article, and it's a real mess. A huge amount needs to be done to it, possibly up to the point of a complete rewrite. Before I criticise it, I should mention that I have just completed a PhD thesis whose topic was a particular strand of neo-logicism. I have also taught some philosophy of mathematics at university level. (I don't mean to boast or wave credentials about. I just want to show that I'm familiar with the material.) I think that this qualifies me to criticise the article.
Anyway, after a read through, here's what I think are some problems with the article:
...
I could go on, but I'm getting tired. It is my opinion that the article could do with a complete rewrite; it is largely very rambling, for the most-part unreferenced, and the notation is completely non-standard. I'm adding a template to this effect now, partly so that people will see my comment here, and hopefully chip in their 2 cents.
I'm happy to have a go at a rewrite. I believe that I am qualified to do so. I'm going to start a version of the article in my sandbox. Then, at the point where I think I have something better than this article (but obviously with ample scope for revisions), I will suggest replacing this article. Would people be opposed to such an idea?
I would suggest a structure for the article as something like the following:
I am happy to have a go at this, and happy to ask for help from others. Again, would anybody be strongly opposed to this? Jdapayne ( talk) 20:35, 7 February 2013 (UTC)
Keep in mind that the article's title is "Logicism", not "Neologicism". So you should be careful to keep your perspective tuned to logicism, especially with regards to the historical; that's why we came to this article, to find out what we could about historical logicism; if we wanted to read about neologicism we'd be reading about it, somewhere or other. I agree with you that neologicism should be split off from this article. Secondly, be sure you have sources for everything. The third comment is a caution: you are apparently new to wikipedia. We've all been newbies, once, and we've all made the same mistakes -- over-enthusiastic, overly critical. My advice to you is to go slow, don't tear everything you see to shreds. At least at the beginning of your efforts, try to find some good in the article and repair the blunders, i.e. work with what's there, build on it . . . slow and steady wins the race. Lastly, keep in mind that a "rewrite" such as the one you are proposing will, someday, be applied to your work, work you will not get an ounce of credit for -- you do not own even the tiniest mote of the work you put into this, and it will get the exactly the same respect you give the work you find here now. Bill Wvbailey ( talk) 02:06, 8 February 2013 (UTC)
You sound like you will be a good editor. Are you familiar with "sandboxing"? I've created dozens of them whenever I want to really have at something, but don't want to clutter the talk page e.g. with quotes, sources, etc. They're fake pages under your "namespace" e.g. the one I used one on this article: User:Wvbailey/Logicism, which might be useful, actually. You would create a new article e.g. User:Jdapayne/Logicism and work it to your heart's content with no need to write "edit summaries" etc.
Here's some background (and some rebuttal) about the article as it is now, in no particular order:
Hope this helps, Bill Wvbailey ( talk) 20:51, 8 February 2013 (UTC)
That does help. Thanks. It was rash of me to suggest a rewrite straight away. What I will do instead is mainly to try and fill in stuff about Frege's logicism - both the philosophical part and the construction - since I think that is something that the article is mainly lacking at the moment, and also the part that I know most about. This might require a bit of reorganising as well to split off the Frege from the Russell bits. I think the idea to separate the construction of the natural numbers out seperately is a good one. I won't try to do that yet, but I'll try to make is so that it's easy to do if needed (by keeping the various bits separate in this article). I've started sandboxes at User:Jpayne/Logicism and User:Jpayne/Logicist constructions of arithmetic for this purpose.
If you're still working on this article, an additional sources that you might want to look at for are the Oxford Handbook of Philosophy of Mathematics and Logic, which has an article on the logicism of Dedekind, Frege. There are also a few articles in the Stanford Encyclopedia of Philosophy which may be useful - especially one on Dedekind, at http://plato.stanford.edu/entries/dedekind-foundations/ , which might help for the Dedekind stuff. I'll have a look at these - and follow up other references - after I've witten some stuff on Frege. Jdapayne ( talk) 10:13, 11 February 2013 (UTC)
Can someone explain what it means for mathematics to be "reducible to logic" (opening sentence)? I'm a mathematician and I have no idea what this means. 158.109.1.23 ( talk) 14:18, 21 February 2008 (UTC)
Is Neo-Logicism meant to be a different article? Because otherwise it references to itself 8/. Fephisto 22:31, 8 July 2006 (UTC)
I am thinking of annihilating this page and rewriting it. Would it be a good idea to move most of the current (not very NPOV) content to a new place, like 'Logicism and Godel's theorem' perhaps? (I'm new here.) -- Toby Woodwark 20:37, 2004 Mar 18 (UTC)
Absolutely not. B 00:15, Apr 26, 2004 (UTC)
I find this page very inaccurate and subjective. If an expert in the area is considering rewriting it I would definitely agree with that.
All right, I'm going in. I don't know if this'll come out perfectly, but I do agree that the second half of the article is extremely subjective, if not downright wrong. I'll try to clean it up now, then...
I removed: "Modern philosophers believed that proof of this theory was the means of banishing the befuddlement...". The idea expressed is probably inaccurate for many of the people usually associated with Logicism.
I also removed: "with sucess except for the paradox of trying to formulate a logical definition of natural numbers in terms of classes". Though it certainly is suspect to use a notion of class to give a logical definition of number, I don't see the sense in which this is a paradox. Wjwma 18:48, 2 August 2005 (UTC)
I modified the sentence alleging that Godel's Incompleteness results undermine logicism. Given certain assumptions and certain formulations of logicism, it is true that the incompletness results undermine logicism, but these positions are controversial. Wjwma 18:21, 3 August 2005 (UTC)
This page is badly in need of some references. Otherwise, it is subject to being deleted per WP:NOR. -- noosphere 09:27, 14 April 2006 (UTC)
I changed the word 'valid' to the word 'alive' in the paragraph on Goedel's Incompleteness Theorems because [especially in a mathematics and logic context] valid means something like 'necessarily truth preserving'. To say that logicism necessarily preserves truth despite being a contested position seems too strong a claim to make in an encyclopedia article. Feel free to change the word to something more appropriate than 'alive' if you can think of something. Taekwandean ( talk) 12:41, 6 January 2011 (UTC)
Just to clarify, I am not saying that the Incompleteness Theorems are not valid, but rather that the philosophical position Logicism may not be. There is quite a lot of literature on that topic, but you might, for starters, take a look at some essays by Thomas Ricketts ( http://www.pitt.edu/~philosop/people/ricketts.html) or Peter Clark ( http://www.st-andrews.ac.uk/philosophy/dept/staffprofiles/?staffid=98) to get a sense of some direct objections. More indirectly, one might think that if the foundations of mathematics are some fundamental intuitions of time and self (something like a Brouwerian kind of intuitionism), then it would certainly not be the case that logicism was necessarily truth preserving (since in this case maths would be reduced to intuitions, not to logic).
But, I think we are likely in agreement, just talking past each other. I am not trying to say that logicism is wrong. I am not trying to say that the incompleteness theorem are invalid. I am, however, trying to draw a distinction between necessarily truth preserving on the one hand, and an active position worth of consideration on the other. So, I still think that the word valid should be changed, but perhaps others could weigh in on this issue? Taekwandean ( talk) 19:45, 7 January 2011 (UTC)
Good, we agree that Goedel does not change the status of the logicism claim. I take it that this is no longer the dispute. What I am trying to change is the wording of the sentence that communicates just this fact that we are not in disagreement about. The word I object to is 'valid'. In most logic and math contexts this word means 'necessarily truth preserving'. It is hard to see how a philosophical position (logicism) that is disputed (by, e.g. the people that I cited among others) could be necessarily truth preserving. This does not mean that logicists are wrong. It just means that in an encyclopedia article we might not want to use such a strong word. But, if you feel strongly that it is necessarily truth preserving, I'll leave it alone. Taekwandean ( talk) 11:37, 8 January 2011 (UTC)
(Unindenting)First, widely held by who? By people who work in foundations or by people who do math in some form? The former are the ones that are important in this context. Second, we are not talking about the math most people do when we are talking about reduced to ZFC, but literally everything that can be termed mathematics; so that's a huge claim. Third, what about large cardinal axioms? If all of mathematics can be reduced to ZFC, then are lca's all undecidable and that's it? That would be a minority opinion. Now, suppose that the Riemann Hypothesis is undecidable using ZFC, but that there is some seemingly unrelated purely set theoretic axiom K so it is true in ZFC + K. If everyone adopted K, would we then start saying that all of math is reducible to ZFCK? What makes ZFC the foundation for all mathematics apart from common convention? My point: ZFC is not implicitly The Foundation, it is a foundation. But this is not the topic at hand, what Frege did with higher order logic and arithmetic reminds me a lot of how arithmetic is done in the Lambda Calculus, the sentence I removed did not seem to be talking about this kind of work. It sounded like it was saying Logicism is valid since things are proved using logic; most of the conversation above seems to be saying this too, this is not Logicism, which is my whole complaint here. If this is not what you mean, then you need to write better. Finally, though it is not directly relevant to the article, you never answered my question about what 'Logic' you are talking about. Do you mean that all of math reduces to first order logic, second order logic, higher order logics all together, some modal logic, etc? Further, this page doesn't really address what a logic is, or what logic in general is; which would be fine, if you weren't talking about all of mathematics (in the philosophical total sense of the term) reducing down to it. I'm not trying to be an ass, but your responses are really missing the point of my objections, I did not come here to attack Logicism, I came here and saw problems with the article and, seemingly, with the people editing it. 71.195.84.120 ( talk) 16:06, 31 July 2011 (UTC)
Intent of these entries: The intent here is to figure out exactly what “Logicism” is and how it came about. I'm only discussing "logicism" here, as it is used in the literature. I am not considering "neologicism".
This will be a work in progress. This is going to take a long time to research. See more at User:Wvbailey/Logicism. .
Philosophy of mathematics: In general what we're dealing with here is Russellian epistomology (see below) with application to the philosophy of mathematics. So we need to look at both the "philosophy of mathematics" and "mathematical philosophy", not at all the same thing. Plus there's Russellian monism; see below.
Mathematics applied to philosophy: If "logic" is to be the core of "mathematics" and "mathematics" is to be the core of philosphy, then "logic" is the core of "philosophy". Thus we have Analytic philosophy. The problem RE philosophy appears to be the epistomological aspect.
The progenitors of logicism: The entry re Analytic philosophy includes Frege, Whitehead and Russell as its progenitors; G-G also includes Schroeder, Dedekind, and Pierce to lesser degrees (G-G 2000:249, see quote under Schroeder below). Plus we have to include the influence of Peano on Russell (G-G 2000:249) by his own admission in his 1901.
Epistemology: see entries below re conflation of Russellian epistemics and mathematics.
Monism, neutral monism: Russell was a neutral monist. See User:Wvbailey/Logicism.
Sources: I'm copying a bit of this from Trovatore's talk page. But to see more, reference in particular Grattan-Guinness 2000:411-506, i.e. Chapter 8 The Influence and Place of Logicism, 1910-1930 and Chapter 9: Postludes: Mathematical Logic and Logicism in the 1930s (pages 507-555), Chapter 10: The Fate of the Search(page 556ff) with various subsections titled 10.1.2 "The timing and origins of Russell's logicism", 10.2 "The Content and Impact of Logicism". His history basically ends with 10.2.3 The fate of logicism.
Most of this will be derived from the huge history in Grattan-Guinness 2000 but I’ve included other sources too. Grattain-Guinness 2000:556 begins his last substantive chapter 10 with this dreary conclusion: "The chapters ends [sic] with a flow-chart for the whole story and some notes on formalism and intuitionism [footnote 1, see below], before locating symbolic logic in mathematics and philosophy in general, and emphasizing the continuing lack of a definitive philosophy of mathematics " [my boldface].
--
Role of Peano, Peano's influence on Russell: TBD [see G-G 2000:250: "It [Schroeder's 1897 paper, his logicism] constrasted greatly with the organic construction of logicism from mathematical logic, already tried by Frege and soon to be adopted by Russell under influence from Peano (§6.5)."
In particular G-G observes that the criticism of Schroeder's 1897 included some important points that "would be of major importance for Russell", and also some changes in notation i.e. substituting iota 'ι' for "the", ⊃ for Peano's Ɔ, 'Cls.' for Peano's 'K'. (G-G 2000:251).
Fregian logicism: TBD
Schroeder's logicism: G-G states here that Schroeder, like Hilbert, wanted a methodology to transform all of mathematics to a routine. Of course this is what Goedel proved could not be done. I've bold-faced this part of the quote:
From Kleene 1952:
From Eves 1990 (notice that he seems to have borrowed from Kleene !):
Livio 2011: In the latest Scientific American article there's an article by Mario Livio "Why Math Works" wherein he discusses two -isms only: Formalism and Platonism (August 2011:81) and tries to answer the question about whether mathematics is intrinsic to the universe and discovered by mankind (Platonism), or whether it is Formalistic in nature -- i.e. invented by mankind. He concludes both seem to be the case.
Livio is a bit perplexed by this universe of ours: "Why are there universal laws of nature at all? Or equivalently: Why is our universe governed by certain symmetries and by locality? I truly do not know the answers . . ." (p. 83).
Unlike Livio, Grattain-Guinness recognizes three mathematical philosophies (footnote 1, page 556: "It is strange that the names for the three main philosophical schools were already in use in ethics (Clauberg and Buislav 1922a, 161). Ethical grounds for exercising the will were 'logicistic' if their consistency was held to be morally sufficient; ethical norms were 'intuitionistic' if they were held to be inborn rather than acquired, and 'formal' if they came through general principles rather than individual objects."
Platonism: As to where Platonism falls in this scheme of things, I have no idea. Perhaps under "logicistic"?
Mancosu on Brouwer's Intuitionism: I found a great quote that corroborates my personal opinion that Logicism is a "practice" rather than a philosophy. Mancosu derives this from Brouwer's 1907 The Foundations. As quoted from Mancosu 1998:9 --
Wittgenstein versus Russell: From G-G's "8.2.5 Russell's initial problems with epistimology, 1911-1912 --
Russell's 1914/1926/1929 "Our knowledge of the external world . . ." :
American criticism of Russell 1914 "Our knowledge of the external world . . .":
Bill Wvbailey ( talk) 00:25, 3 August 2011 (UTC)
Annotated Bibliography (Sources):
In the first section, the article says ' theorems of mathematical logic [...] can be proven using the fundamental theorem of arithmetic (see Gödel numbering) '. A reader would be forgiven for taking that to mean that the crux (cruces?) of the proofs of many theorems of mathematical logic is an appeal to the Fundamental Theorem of Arithmetic. Which is of course not the case.
I suspect what the author had in mind is how, in its setup phase, Gödel's proof of his First Incompleteness Theorem uses a simple construction with primes and exponents to in principle assign a natural number to any well-formed formula (wff) in the formal language. The Fundamental Theorem of Arithmetic guarantees the uniqueness of this assignment: if a given natural number corresponds to any wff at all, then it corresponds to exactly one of them.
The result of this construction is of course used in the deep part of the proof, when one feeds the Gödel Sentence its own assigned number through its free variable, but the proof in no way follows directly from the Fundamental Theorem of Arithmetic.
Would the author please reword his statement in order to prevent such unwarranted interpretation of it. The fact that each whole number has a unique prime factorization is in no way a silver bullet to help us prove theorems of mathematical logic. — Preceding unsigned comment added by Schumacher peter ( talk • contribs) 17:46, 13 September 2016 (UTC)
---
This article makes some pretty strong claims for an encyclopedia. First of all, that the incompleteness theorem undermines logicism because it shows that there are undecidable statements is a very subjective position and I think this should be emphasized. The fact that some propositions can neither shown to be true nor false within a particular deductive system does not impede the reduction (or rather translation) of informal mathematics into formal deductions.
Second, the sentence "Therefore, any claim that logicism remains a valid concept must strictly rely on the dubious notion that a system of proof based on man-made models is precisely as powerful and authoritative as one based on the existence and properties of the natural numbers." does not make any sense. I have yet to see any mathematics that is not man-made (proof assistants are man made as well of course). The sentence also seems to claim that the existence and properties of natural numbers are not man made, which might be true from a Platonist point of view, but I doubt that this would be the position of adherers of logicism. This sentence seems like an ill-founded, biased statement about logicism.
The strong position against logicism taken by this article in the introduction of an article does not seem to follow the principles outlined by wikipedias neutral point of view. /info/en/?search=Wikipedia:Neutral_point_of_view — Preceding unsigned comment added by 2601:580:8205:183F:ED7E:783:7B77:B04A ( talk) 21:57, 8 March 2017 (UTC)
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==missing in Wiktionary==
=logicalism=
<code>Wiktionary code</code>
==English==
===Etymology===
{{suffix|en|logical|ism}}
===Noun===
{{en-noun|~}}
# {{lb|en|philosophy}} The doctrine that any possible real world has logical foundations.
====Synonyms====
* {{sense|philosophy}} {{l|en|metaphysical logicism}}
* {{sense|philosophy}} {{l|en|materialism}}
* {{sense|philosophy}} {{l|en|philosophical materialism}}
* {{sense|philosophy}} {{l|en|physicalism}}
* {{sense|philosophy}} {{l|en|metaphysical naturalism}}
====Antonyms====
* {{sense|philosophy}} {{l|en|idealism}}
====Hypernyms====
* {{sense|philosophy}} {{l|en|monism}}
* {{sense|philosophy}} {{l|en|realism}}
====Translations====
{{trans-top|philosophical position}}
* Finnish: {{t|fi|logikalismi}}, {{t|fi|metafyysinen logismi}}
* Galician: {{t|gl|logicalismo|m}}
* Greek: {{t|el|μεταφυσικός λογικισμός|m}}
* Italian: {{t|it|logicismo metafisico|m}}, {{t|it|logicalismo|m}}
{{trans-bottom}}
====See also====
* {{sense|philosophy}} {{l|en|dualism}}
====See also====
* {{l|en|idealism}}
The fact that there's no [omni]universal axiomatics = no omniaxiomatics = there isn't the axiomatic system which includes all the infinite possible (even the infinite weird experimental ones we can make) axiomatic systems, [that fact] is very important and fundamentally related both in logicism (mathematical logicism) and in logicalism (metaphysical logicism).
2A02:2149:8B63:D400:BD1F:56E8:B586:E393 ( talk) 03:49, 9 November 2023 (UTC)
Some neologicists claim that the axiomatic system of all axiomatic systems (omniaxiomaticity/omniaxiomatics/universal axiomatic system) doesn't exist because it would include mutually exclusive axiomatic systems; and the set of all sets doesn't exist (thus it cannot be a set).
Some neologicists/ modern logicists and other mathematicians work on experimental axiomatic systems. Infinite axiomatic systems are logically possible. The vast majority of logically possible axiomatic systems aren't particularly useful. Mathematics is a proof system (see: John Stillwell). Allomathematics is mathematics based on different axiomatics. A true allomathematics must have some mathematical use, for example it can be better than common mathematics for particular types of proofs, computer chip building, etc. Physics isn't a proof system but a substantiality system, and the axiomatic prerequisites for physical foundations/ of the physical foundations (in US English) include a more linked axiomaticity = axiomatic foundations (program-based axiomatics and not a list in order the physical foundations are a specific entity kernel and not diffused axioms logically unlinked that they could be otherwise thus cannot make solid physical foundations). Physical axiomatics requires incorporated the forms of entropy (thermodynamic entropy and informational entropy) otherwise we have timeless stationary geometry and not spacetime. Physical axiomatics isn't based on crystal clear axioms but on engaged axioms with other axioms and with necessary endosystemic interpretations. Quantum mechanics meets the prerequisites for physical foundations, thus the quantum foundations is quantum mechanics. It's supposed vagueness is mechanisms of engagement of other procedures and interpretations because actuality is always relational. The self-causation/ self-causality criterion is met by a logical axiomatic program kernel which is self-caused due to being logical and the quantum foundations isn't tautological to the Big Bang. The Big Bang is a very important event but it's not the quantum foundations. In cyclic cosmology infinite big bangs exist; the universe expands till a big rip so mass–energy producing which collapses into black hole center substance = maximal degeneracy chromodynamic superfluid which fragments into inflation particles and normal big bang (homogenous and isotropic) begins (immediate big rips don't produce homogenous and isotropic big bangs and violate the axiomatic prerequisites for physical foundations). (Versions of cyclic cosmology are mere guesses but they are important to understand the difference between big bang = important event and the logical foundations of substantiality). Not one foundations of logic exist. The ideal Turing machine should have the infinite transcribers for different axiomatic systems. Some axiomatic systems are wholly transcribable, and others in a case-by-case manner. Mathematics can describe all axiomatic systems but many axiomatic systems aren't flexible enough to be able to describe others. Thus many axiomatic systems aren't transcribable to others. The idealized Turing machine isn't supposed to merely calculate simple things but it should be also able to solve difficult mathematical problems which require non-self-evident techniques (otherwise we don't have a true Turing machine because it fails in some categories of calculations). The Turing machine is impossible to be locally realised because the required mathematical techniques are infinite, and infinity is never realisable because in the axiomatic prerequisites for physical foundations infinity isn't relational to specific procedures in a manner which leads to substantiality = existence. Existence cannot ever be infinite. Existence can be linked to infinite phenomena but it cannot be infinite in itself. — Preceding unsigned comment added by 2A02:2149:8B83:6500:7813:142C:413A:4290 ( talk) 09:15, 11 March 2024 (UTC)
In modern forms of logicism infinite different logical systems are possible. Not all logical systems have to be proof systems like mathematics (see: John Stillwell on proof).
Comparative logicism compares the traditional logicism, the forms of neologicism including the very weird experimental non-proof logicisms (infinite are possible).
Comparative logicism is important because "some ways to build parts of logical systems are more effective than others". Thus we can say that traditional logicism is wrong for being very idealistic, but comparing elements of various forms of logicism remains important because at least partially some logical foundations seem more effective than weird experimental logical foundations. It's something like the Feynman diagrams. We can create very needlessly weird logical foundations, but some more traditional logical foundations are more probable. But a physical foundations of any universe would include some more probable and other less probable foundational parts (comparative logicism understands that all forms of logicism can be part of the axiomatic foundations of universes = physical laws). — Preceding unsigned comment added by 2A02:2149:8BE9:C00:F0A3:E95F:3E9E:9FBF ( talk) 17:57, 13 April 2024 (UTC)