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A request for referencing[edit]

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 (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 (talk) 16:16, 3 March 2015 (UTC)

This article is a mess. I suggest a rewrite[edit]

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:

  • The very first statement is unclear: mathematics being an extension of logic and mathematics being reducible to logic (and definitions) are very different things. The second -- which is what is more usually called 'logicism' -- is much more controversial. As it happens, the reference attached to this sentence no longer exists. A better reference would be, for example, the Stanford Encyclopedia of Philosophy.
  • The first paragraph is very long winded and unclear. It is also almost entirely unreferenced. I think it would be better to have a brief introduction before the contents (saying roughly what logicism is, and who has propounded it). The kinds of details in this paragraph should then go in appropriate points in the article (e.g. a section on history, a section on criticisms and so on).
  • The stuff about Gödel in this section is also well off-mark. 'This theorem is proved with logic just like other theorems' is more-or-less nonsense. In one sense, it is absolutely false - Gödel's theorems have to be made in the background of a mathematical theory, and is not a theorem of pure logic. That logic is used on top of this mathematical theory is irrelevant, since logicism is far more than the (uncontroversial) claim that mathematics uses logic. In any case, even if it were claimed that the relevant mathematical theories can be reduced to pure logic in some sense, this would not by itself invalidate it as a criticism; it may then just show that logicism is self-undermining. Most importantly, however, I am unaware of anybody making such a rebuttle in reputable published work. As such, the stuff on Gödel here constitutes original research. (There are published rebuttals of this kind of objection, and I think they deserve mentioning, but in a section dedicated to criticisms.)
  • I don't claim to know much about the origins of the term 'logicism'. On the face of it, however, most of the discussion in this section does not address the point, but talks about the origin of other words with 'logic' as their root.
  • The stuff about symbolic logic is close to nonsense. What has algebraic logic got to do with anything? And algebraic logic is a different thing from Boolean logic. The discussion of what symbolic logic is is itself very unclear. What might be useful is a quick characterisation of predicate, as opposed to propositional logic.
  • The stuff on history is rambling and unclear

Epistemology section[edit]

  • It's really unclear what the author of this section means by 'epistemology'. Most of what's discussed has nothing to do with epistemology. Realism and idealism aren't epistemological positions, for example. Most of it is metaphysics, and very rambling at that.
  • Russell's paradox definitely does not belong in this section, since it prima facie has nothing to do with epistemology. There are a few errors in it as well. The paradox was not in Begriffsschrift, but in Grundgesetze
  • Generally, this section is incredibly rambling. It is not easy to follow, and it is hard to see how it fits together. Most of the stuff doesn't belong in a section entitled 'Epistemology', but in its own section.

The Logicistic construction of the natural numbers[edit]

  • Firstly, the word 'logicistic' is really not right. It should be 'logicist'.
  • The axiom of pairing in set theory really has nothing directly to do with the usual interpretations of arithmetic in set theory. Ordered pairs especially, as far as I remember, are not directly used in the construction of von-Neumann ordinals, nor (obviously) Zermelo numerals
  • The stuff about types is completely the wrong way around. Modern set theory is untyped, and so the numbers (i.e. the von Neuman ordinals or Zermelo numerals) are all of the same type. By contrast, the issue of types does come up in logicist foundations of arithmetic, since some of these (especially Russell and Whitehead's) are contructed in a type theory. Their system results in a collection of numbers at every level of the type-theoretic heirarchy.
  • As with the previous sections, this is very rambling. Sentences follow on from one another without it being at all clear how they fit together.


  • It's very unclear what is meant by classes 'coming about as the result of propositions'.
  • Quoting chunks of Russell doesn't make stuff very clear. It would be better to give a beginner-friendly introduction of the material that's necessary to understand the rest of the article. As it happens, most of these terms are not really needed.

The definition of the natural numbers[edit]

  • the first sentence of this makes no sense. It is also written in a very much unencyclopic manner.
  • 'devise' is the wrong word.
  • the example is very unclear. It does not illuminate the quotation at all. Its relevance is questionable.
  • The notation of | a , b ,c | is completely non-standard. It should instead be {a,b,c}.
  • The 'bundles' example is wrong. As it is it suggests that the displayed sets are equivalence classes under equinumerosity. They are not. The equivalence class of pairs features all pairs, for example.
  • 'This peculiar non-existent entity' doesn't make sense, at least, not without a substantive discussion of Russell's (and Meinong's!) views on a distinction between existence and subsistence which does not belong here.
  • The use of non-standard symbols in entirely unhelpful.


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:

  • brief introduction, saying what logicism is
  • A brief history
  • Pre-Frege logicism - the most notable forms of logicism are those deriving from Frege and Russell, so it seems reasonable to lump pre-Frege stuff together
  • Frege's logisim, divided into a number of sections:
    • Frege's motivation and philosophical views about logicism
    • Frege's technical programme as carried out informally in the Grundlagen and formally in the Grundgesetze
    • Russell's paradox and the failure of Frege's programme
  • Russell and Whitehead's logicism. Again split into multiple section. This is not my area of expertise, so I will to some extent leave this as a skeleton for those more knowledgeable than me to improve.
  • Criticisms
    • I am unsure whether it is better to have a by itself for criticisms, or for each section to have a subsection -or citicisms.
  • Neo-logicism.
    • This section could do with expanding. I think that I will also have a go at creating an article specifically for neo-logicism. It is a major topic in contemporary philosophy of mathematics, so I believe that it is notable enough.

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)

Jdapayne criticism[edit]

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)

Thanks for the advice.
  1. I understand that the article is about logicism. I mention neo-logicism because (a) it is itself a branch of logicism and (b) since neo-logicism is very much inspired by historical logicism, my own work on neo-logicism means that I have a good understanding of historical logicism. I don't intend to make the article about neo-logicism, any more than it it already. The section on neo-logicism is probably fine as it is.
  2. I will indeed make sure I have sources for everything. In fact I think that one of the main problems with the current article is that it makes a lot of bold statements in places which are not referenced. I am very used to citing sources meticulously in the writing that I do, and editting Wikipedia will be no exception.
  3. Understood completely. I'm not doing this because I want credit or anything like that. The reason I want to rewrite (or nearly rewrite) this article is because I want Wikipedia to be better. It has been a great resource for me in the past and I want to put something back. It's also because I don't want students of mine coming to this page as it is for additional reading during a lecture course of mine in the future; in its present state it is likely to confuse at best, and at worst impart a lot of incorrect information. Likewise, I'm completely open to (and used to) criticism. If what I write gets steamrollered because somebody thinks that it can be improved, then that is fine. Again, the ultimate aim is to make Wikipedia a better resource for everybody. On the 'go-slow' advice: thanks. Maybe what I'll try doing first is reorganising the current content, rather than a rewrite - one of the main issue with it at the moment is that it's not very well structured. That then makes it possible to modify in smaller chunks. Jdapayne (talk) 16:27, 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:

  • About the strange symbolism, the vertical bars, for the example:
┊a, b, c┊, ┊d┊, ┊┊, ┊e, f, g┊, ┊h, i┊, ┊j, k┊, ┊l, m, n, o, p┊, ┊q, r┊, ┊s┊, ┊t, u┊, ┊v, w, x, y, z┊
The idea is to communicate that these are not sets (classes) in the conventional symbolism of { }. This is in light of Russell's "no-class" theory, and in particular Goedel 1944's criticism.
  • Logicistic versus logistic or whatever: I've seen it written a number of ways.
  • The matter of "epistemology", maybe this is the wrong word -- ontology? [cf article on Meinong], or just "philosophy?". Whatever, this section and the criticism, I believe are the most important sections. At historical stake were the arguments re mathematical "foundations" -- Platonism, Logicism, Formalism, Intuitionism, constructivism, "mathematical realism" (cf Goedel 1944:106ff, 120ff). I coudn't find much at all about the philosophies of the Logicists and had to go back into primary sources e.g. Dedekind, Frege and Russell 1903 and 1919. The question is: "How do we come by the 'knowing of [the concept of] number' " or "Why do we know how to count things"? Are numbers "Platonic entites" that exist out there all by their little lonesomes? Are they then discovered? Or, are they invented? Are they "free creations of the human mind"? Or, are they a priori (built-in) into our genetics, a tendency we have because of the structures of our brains? (Dedekind) I have a quote from Bernays (in Mancosu I believe) and it's similar to the Kleene quote having to do with where we derive our "number sense" -- from an intuition of sequence. This is what's at stake in this section.
  • Sourcing -- my best secondary critical sources was Goedel 1944. My secondary sources were the commentaries to be found ibefore the appropriate articles in van Heijenoort, commentary before Goedel 1944 (Charles Parsons), Mancosu, and Grattain-Guiness, and the bit of commentary before Russell 1919 (Hagar's commentary). Unfortunately I had to use primary sources (Russell 1903, Russell 1919, Frege, Dedekind, and Peano). Dedekind bothered me because I wanted to know more about his "philosophy of number". I don't have the language skills (can't read German) to go sleuthing about in ancient primary sources, and that sort of behavior can get you into O.R. difficulties, anyway. I had to go into Russell 1903 to see what was going on with his "unit class" misery, and his 1919 to figure out what his "philosophy of number" was. And that's pretty much the extent of the sourcing. If others e.g. Meinong are not mentioned it's because I know zip about them.
  • One thought would be to break out the example of the Logicist construction of the natural numbers into its own sub-article. That might "densify" ("compactify") the article.

Hope this helps, BillWvbailey (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 , 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. (talk) 14:18, 21 February 2008 (UTC)

Being a reckless fellow, I'll have a go :-). Suppose a 'logic' is a collection of symbols in terms of which formulae are defined, some of those formulae are deemed axioms, and yet others (derived from those axioms using rules of inference) are deemed theorems. Now if all the symbols of mathematics can be defined in terms of the symbols of that logic, then all formulae of mathematics become formulae of that logic. And if, further, all the true formulae of mathematics (or, at least, all those known to be true to date) are among the theorems of that logic; then mathematics has been reduced to that logic.
'But' I hear you ask, 'what is a logic?' Good question. If the above programme were carried out today, I guess ZFC would be used. But many people would say that the axiom of infinity goes beyond logic, so ZFC wouldn't count as a logic. And, in Russell's type theory, reducibility as well as infinity would rule it out.
Let's hope that someone who really knows the answer comes along and corrects that. Meanwhile see what Carnap says here: Philosophy_of_mathematics#Logicism (talk) 01:49, 7 November 2014 (UTC)


Is Neo-Logicism meant to be a different article? Because otherwise it references to itself 8/. Fephisto 22:31, 8 July 2006 (UTC)

Sometimes parts of large articles refer to other parts of the article. Nevertheless, the problem is fixed. —The preceding unsigned comment was added by Canadianism (talkcontribs) 19:07, 28 February 2007 (UTC). Canadianism 19:07, 28 February 2007 (UTC)

Should this page be moved and replaced?[edit]

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...

sentences removed[edit]

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)

The incompleteness results[edit]

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)

I'm no expert, but I believe Incompleteness just undermine formalism; although Frege developed a formal system for his work, and later Russell did in Principia, I think the formalism used is inessential to the ideas underneath it. Whatever this may be, I think Incompleteness, if not shown to actually undermine formalism should be deleted. If I'm completely wrong and Incompleteness really reveales fundamental limitations to logicism, then I think it should be added an entry to the contents. Sergio89 (talk) 01:19, 18 August 2012 (UTC)

Citations please[edit]

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)

Good enough yet? Canadianism 18:49, 28 February 2007 (UTC)
This page is in no danger of being deleted "per WP:NOR." The topic of logicism is quite notable and so would survive any deletion discussion. If you believe that the page needs additional references, fix it, but please don't make threats about deletion. CMummert · talk 18:56, 28 February 2007 (UTC)
I think, more or less, it has been fixed. —The preceding unsigned comment was added by Canadianism (talkcontribs) 19:05, 28 February 2007 (UTC).
See also Philosophy_of_mathematics#Logicism which is somewhat more detailed than this article. —Preceding unsigned comment added by (talk) 07:11, 21 September 2007 (UTC)

Small Change[edit]

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)

You say that this is a contested issue, but I doubt that. Can you cite some authority who says that it is not valid? Roger (talk) 16:06, 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 ( or Peter Clark ( 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)

I looked at their web pages, and they have no essays there. I am not sure why their opinion would matter anyway, as they are philosophers. The claim is that math is reducible to logic, and Goedel does not change that. Roger (talk) 21:13, 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)

The phrase "the basic spirit of logicism remains alive" seems preferable since mathematics, with all its symbols, falls short of machine code. A glance at the troubles in line (geometry) will show primitive notions continue to crop up as we try to tie down mathematics. The state of the logicism thesis might be treated separately from Goedel's result.Rgdboer (talk) 19:52, 8 January 2011 (UTC)
" I am not sure why their opinion would matter anyway, as they are philosophers." Really? This is an article about the philosophy of mathematics, not mathematics. What does it mean, mathematically, to say that mathematics reduces to logic? Not much since there is no mathematically formal definition of mathematics. Anything about mathematics as an entire domain is going to be philosophy, so your point doesn't make any sense.
"To say that logicism necessarily preserves truth" this doesn't make any sense. Logicism is a position in the philosophy of mathematics, it is not an inference, logicism does not act on anything nor is it an argument structure, etc. etc. In short, valid would not be referening to validity in the way you seem to be using it...
Honestly, not to be rude, but do either of you know what, and I mean really know, what you are talking about? For example, what does, "The phrase "the basic spirit of logicism remains alive" seems preferable since mathematics, with all its symbols, falls short of machine code. A glance at the troubles in line (geometry) will show primitive notions continue to crop up as we try to tie down mathematics. The state of the logicism thesis might be treated separately from Goedel's result." mean? It sounds kind of meaningful, but its just fluff and non sequiter..."mathematics, with all its symbols, falls short of machine code."? This is either just supposed to sound cool or it's poorly expressing one of the many things it could vaguely mean, most non topical. Etc.Etc.Etc. Sorry, this just ticks me off, you are editing and debating a(n) articl/topic that you clearly are uninformed about, which wouldn't be such a big deal, if it wasn't in a public resource and about a topic most people are already lacking general knowledge on. (talk) 09:46, 29 July 2011 (UTC)
It appears to me that you deleted good material just because you disagree with logicism. You delete one sentence because it is a "strong statement". Is it true or not? Roger (talk) 14:27, 29 July 2011 (UTC)
I removed, "However, the basic spirit of logicism remains valid, as that theorem is proved with logic just like other theorems". Logicism is about mathematics being reduced to logic, not about logic being used in proofs, so this sentence isn't even about the article's topic; or you are asserting something really strong, which I doubt, and would require a lot more than what you wrote to state. I also removed, "Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of Zermelo-Fraenkel set theory (or one of its extensions, such as ZFC), which has no known inconsistencies (although it remains possible that inconsistencies in it may still be discovered). Thus to some extent Dedekind's project was proved viable, but in the process the theory of sets and mappings came to be regarded as transcending pure logic."
First, even if all of "Mathematics" can be reduced to ZFC- which a way way stronger statement than you think (What about ZF, what about ZFC + GCH, what about intuitionism, what about category theory, etc?) and more controversial than you think- But even if that were 100% a meaningful and true statement, it would not imply logicism. Set Theory is not Logic, it is foundational, but that isn't the same thing. Set Theory is about Sets. For an analogy: Church's Lambda Calculus can represent natural numbers and simulate arithmetic, normal arithmetic uses recursive ideas; this does entail that Arithmetic is really just recursion. Now, you are applying the same logic, except you are replacing arithmetic with all of mathematic and employing the much more dubious claim that everything reduces to ZFC and, the even more unjustified claim that Logic(which one exactly?) contains a reduction of ZFC. My point: you are either stating something really really controversial and strong without justification, or you are confused as to what that statement means. I'm going to guess the latter since in half of the above conversation you guys were having, no one seemed to understand that logicism was philosophy and not mathematics. Finally, I have nothing against logicism, I do have something against people writing an encyclopedia article on a topic when they don't seem to understand it; again, this is an encyclopedia article, you need to really understand what you are writing about and all of its implications. (talk) 05:17, 30 July 2011 (UTC)
By the way, I do realize what I removed contains, "but in the process the theory of sets and mappings came to be regarded as transcending pure logic." And that this seems to counter my whole complaint about ZFC not being logic. However, this seems tacked on and obscures that very point, moreover, it is still assuming a lot about the role of ZFC and I'm not sure that anyone ever considered ZFC as logic, which makes the whole 'in the process' and 'transcended' stuff seem like bad wording. Just preempting:-) (talk) 05:21, 30 July 2011 (UTC)
So you might accept that math is reducible to ZFC, but you don't see what that has to do with reducing math to logic. Is that correct? Roger (talk) 06:20, 30 July 2011 (UTC)
I don't accept that mathematics is reducible to ZFC. ZFC is not logic, it is set theory, so yeah, I don't see what that had to do with it. I don't accept that ZFC is Set Theory, but that it is a theory of sets. Finally, what are you talking about when you say reduce to logic? Do you mean that it can all be carried out in some logic, presumably classical? Do you mean that all of mathematics can be carried out using first order logic? (These are very different by the way...) Do you mean it in the sense of what Frege did with higher order logics and arithmetic? What about Topoi and other category theoretic notions? What about set theories that reject choice, or even have something that contradict it? What about set theories that reject the axiom of foundation? What about computation theory? Etc.Etc.Etc. My point: you aren't saying anything that meaningful. In fact, the article and the discussion seemed to be a confused jumble of terms someone picked up as an undergrad skimming through books...which is fine, if it wasn't an encyclopedia article. (talk) 10:40, 30 July 2011 (UTC)
Yes, I do think that all of math can be reducible to logic, and that it can all be formalized in ZFC. That includes topoi and computation theory, altho that takes some extra work. I also say that this is a widely held view, altho followers of intuitionism and other ideas may disagree. But my personal opinion is irrelevant. You disagree with logicism so you have removed sentences that describe logicism. That is wrong. You can say that logicism is not accepted by everyone and give links to articles describing other views, but please do not delete sentences describing logicism from a logicism article. Roger (talk) 14:47, 30 July 2011 (UTC)
Amen Greg Bard (talk) 16:08, 30 July 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. (talk) 16:06, 31 July 2011 (UTC)

You're right, I don't really get your point. The article is about logicism, and you seem to want to discuss other issues. I restored the deleted sentences, because they seem correct to me. Do you have some reliable source that says otherwise? Roger (talk) 22:36, 1 August 2011 (UTC)
Wow! I mention one thing as not being on topic out of everything I said and you dismiss everything I said. The sentence saying that Logicism's spirit is still viable because the incompleteness theorem was proved using logic is garbage, the discussion at that point is about the implications of it, not how it was arrived at. If Godel's Theorem deals a blow to logicism, then it does; even if it is in the spirit of logicism. You say they seem right to you? This page offers a very weak dissent with the mention of Godel and ZFC, but immediately says that everything is all okay because some logic was used, nothing is gotten into, just swept under the rug. Not to mention that these sentences read like they were tacked on to defend someone's personal beliefs; no arguments, just bald assertion. That aside, you don't even get into the meat of Logicism, you don't discuss any of the variants of it. Finally, I go back to this, the initial conversation in this topic sounds like the editors involved don't know the topic at hand. (talk) 07:16, 2 August 2011 (UTC)
I do say that logicism remains valid. It is not garbage. If you disagree, then you ought to be able to point to some mathematics that is not reducible to logic. Variants of logicism are mentioned under "neo-logicism". Do you have any suggestions for improving the explanation of logicism? Roger (talk) 16:27, 2 August 2011 (UTC)

History of Logicism, including historical criticism and developments[edit]

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:

". . . [Schroeder] went through an intellectual conversion under the influence of [Frege's] logic. In §4.4. we saw that his Vorlesungen were broadly Boolean, in that mathematics wa used to analyse logic; but after completing his volume on the logic or relatives he reversed these roles. He publicised his change in a paper delivered to the International Congress of Mathematicians of 1897 . . .: 'I may incidentally say, that pure mathematics seems to me merely a branch of general logic' (1898a, 149). Dedekind's cryptic claim that arithmetic was part of logic . . . was one inspiration; Pierce's logic of relatives was adduced as a more specific one, since it provided means of expressing all the basic 'categories' for matchematics . . ..¶ . . . He proposed an 'absolute algebra', in which algorithmic methods would be applied to the algebra of logic to turn out all possible combinations of relation, connective and propostion, together with the laws appropriate to each case; then the entirety of a mathematics was to be cumulatively delivered, case after case. This cataloguing approach to logicism rather resembled his extensional onception of cases . . .; it contrasted 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)."(G-G 2000:249).

From Kleene 1952:

"The logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the intuitionistic view, an essential mathematical kernel is contained in the idea of iteration, which must be used e.g. in describing the hierarchy of types or the notion of a deduction from given premises. || Recent work in the logicistic school is that of Quine 1940. A critical but sympathetic discussion of the logicistic order of ideas is given by Goedel 1944." (Kleene 1952:46)

From Eves 1990 (notice that he seems to have borrowed from Kleene !):

"Whether or not the logistic thesis has been established seems to be a matter of opinion. Although some accept the program as satisfactory, others have found many objections to it. For one thing the logistic thesis can be questioned on the ground that the systematic development of logic (as of any organized study) presupposes mathematical ideas in its formulation, such as the fundamental ideas of iteration that must be used, for example, in describing the theory of types or the idea of deduction from given premises." (Eves 1990:268).

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 --

" 'The Foundations' (B1907) defines "theoretical logic" as an application of mathematics, the result of the "mathematical viewing" of a mathematical record, seeing a certain regularity in the symbolic representation: "People who want to view everything mathematically have done this also with the language of mathematics . . .the resulting science is theoretical logic . . . an empirical science and an application of mathematics . . . to be classed under ethnography rather than psychology" (p. 129) || The classical laws or principles of logic are part of this observed regularity; they are derived from the post factum record of mathematical constructions. To interpret an instance of "lawlike behavior" in a genuine mathematical account as an application of logic or logical principles is "like considering the human body to be an application of the cience of anatomy" (p. 130).

Wittgenstein versus Russell: From G-G's "8.2.5 Russell's initial problems with epistimology, 1911-1912 --

"Clearly Russell's old and new philosophical concerns were involved, since the epistemology of logic was a major issue. One criticism was that Russell's theory could not handle asymmetrical relations and discriminate between 'S believes that a precedes b' and 'S believes that b precedes a'. . . . || Another dart was aimed at Russell's logic, where already in June 1912 Wittgenstein was convinced that 'The prop[osition]s of Logic contain ONLY APPARENT varibables' so that 'there are NO logical constants' (presumably meaning that they were not objects: Letters, 10). He must have realised that Russell's logic was muddled up with logicism and so needed its own characterization; the may have been led to his view of variables by noting that a logical order was specified by its quantified variables. he seems to have accepted type theory; but he did not subscribe to logicism, since he concluded this letter tha 'Logic must turn out to be of a TOTALLY different kind than any other science', presumably including mathematics. || For logicist Russell, however, such issues were still more serious, and confidence in his new book gradually disintegrated . . . However, the general thrust of his philosophy -- empiricist and reductionist epistemology drawing upon techniques from logic -- was unimpaired . . .." (G-G 2000:481-482)

Russell's 1914/1926/1929 "Our knowledge of the external world . . ." :

"In an opening chapter he summarized the prevailing philosophies, idealism and evolutionism, which he wished to replace: Bradley and henri Bergson were the respective prime targets. The positive doctrine was displayed across six chapters, prefaced by his creed of 'Logic as the essence of philosophy' (ch. 2) . . . Then in ch. 3 using those first six words [of the full title], he summarized a version of knowledge by acquaintance, although he neither used that name nor analyzed judgment or truth-values of propositions: the failed book of the previous year seems to have reduced his ambitions. Overall the book is a patchy scenario, more mathematics than necessary and not really enough science. But the important role of logic was clear; and while he avoided a symbolic treatment, the fusion of logic with epistemology was to give the book a warm reception . . ." (G-G 2000:423).

American criticism of Russell 1914 "Our knowledge of the external world . . .":

"[Theodore de Laguna] . . . while praising the merits of the method ['the logical-analytic method in philosophy'] in its mathematical contexts (1915a, 451) he greatly doubted its utility in Russell's new book, stressing the epistemic dangers of using Ockham's razor (p. 453) and sensing a vicious circle in constructing space via perspective spaces: 'Mr. Russell has deduced his conclusion from his knowledge of physical space: nobody ever induced it' (p. 462). He concluded that 'Mr. Russell's philosophy is as complete and radical a failure' as a theory of ethics which Russell had recently abandoned (p. 462).
"Logicism was discussed also in mathematical journals . . . One of the best reviews of the first volume of PM was 25 pages written by J.B. Shaw in its Bulletin. . . . [here] he stressed how little mathematics seemed to fall with the logicist purview. . . . || In a sequel paper in The monist Shaw 1916a continued in the same vein, concluding that 'Logistic hs a right therefore to exist as an independent branch of mathematics, but is not the Overlord of the mathematical world' (p. 414)" (G-G 2000:421ff "8.3 Logicism and Epistemology in America and With Russell, 1914-1921".

Bill Wvbailey (talk) 00:25, 3 August 2011 (UTC)

Annotated Bibliography (Sources):

  • I. Grattan-Guinness, 2000, The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and The Foundations of Mathematics from Cantor Through Russell to Gödel, Princiton University Press, Princeton NJ, ISBN 0-691-05858-X.
  • Paolo Mancosu, 1998, From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, New York, NY, ISBN 0-19509632-0.
  • Howard Eves, 1990, Foundations andFundamental Concepts of Mathematics Third Edition, Dover Publications, Inc, Mineola, NY, ISBN 0-486-69609-X.
  • Stephen C. Kleene, 1971, 1952, Introduction To Metamathematics 1991 10th impression,, North-Holland Publishing Company, Amsterdam, NY, ISBN: 0 7204 2103 9.
  • Mario Livio August 2011 "Why Math Works: Is math invented or discovered? A leading astrophysicist suggests that the answer to the millennia-old question is both", Scientific American (ISSN 0036-8733), Volume 305, Number 2, August 2011, Scientific American division of Nature America, Inc, New York, NY.
  • Bertrand Russell, 1903, The Principles of Mathematics Vol. I, Cambridge: at the University Press, Cambridge, UK. In particular the Preface pages vi-ix wherein Russell asserts:
  • First: "the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles",
  • Secondly: "the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task . . . The discussion of indefinables-which forms the chief part of philosophical logic -is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple."
  • Bertrand Russell, 1919, Introduction to Mathematical Philosophy, Barnes & Noble, Inc, New York, NY, eISBN: 1-4113-2942-7. This is a non-mathematical companion to Principia Mathematica.
    • Amit Hagar 2005 Introduction to Bertrand Russell, 1919, Introduction to Mathematical Philosophy, Barnes & Noble, Inc, New York, NY, eISBN: 1-4113-2942-7.

This Assertion is Potentially Highly Misleading[edit]

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 (talkcontribs) 17:46, 13 September 2016 (UTC)


On undecidability and neutrality[edit]

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. — Preceding unsigned comment added by 2601:580:8205:183F:ED7E:783:7B77:B04A (talk) 21:57, 8 March 2017 (UTC)

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