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Philosophical analysis

Philosophical analysis is any of various techniques, typically used by philosophers in the analytic tradition, in order to "break down" (i.e. analyze) philosophical issues. Arguably the most prominent of these techniques is the analysis of concepts (known as conceptual analysis).

Method of analysis
While analysis is characteristic of the analytic tradition in philosophy, what is to be analyzed (the analysandum) often varies. In their papers, philosophers may focus on different areas. One might analyze linguistic phenomena such as sentences, or psychological phenomena such as sense data. However, arguably the most prominent analyses are written on concepts or propositions and are known as conceptual analysis (Foley 1996).

A.C. Ewing distinguished between two forms of philosophical analysis. The first is "what the persons who make a certain statement usually intend to assert" and the second "the qualities, relations and species of continuants mentioned in the statement". As an illustration he takes the statement "I see a tree", this statement could be analysed in terms what the everyday person intends what they say this or it could be analysed metaphysically by asserting representationalism.

Conceptual analysis consists primarily in breaking down or analyzing concepts into their constituent parts in order to gain knowledge or a better understanding of a particular philosophical issue in which the concept is involved (Beaney 2003). For example, the problem of free will in philosophy involves various key concepts, including the concepts of freedom, moral responsibility, determinism, ability, etc. The method of conceptual analysis tends to approach such a problem by breaking down the key concepts pertaining to the problem and seeing how they interact. Thus, in the long-standing debate on whether free will is compatible with the doctrine of determinism, several philosophers have proposed analyses of the relevant concepts to argue for either compatibilism or incompatibilism.

A famous example of conceptual analysis at its best is given by Bertrand Russell in his theory of descriptions. Russell attempted to analyze propositions that involved definite descriptions (such as "The tallest spy"), which pick out a unique individual, and indefinite descriptions (such as "a spy"), which pick out a set of individuals. Take Russell's analysis of definite descriptions as an example. Superficially, definite descriptions have the standard subject-predicate form of a proposition. For example, "The present king of France is bald" appears to be predicating "baldness" of the subject "the present king of France". However, Russell noted that this is problematic, because there is no present king of France (France is no longer a monarchy). Normally, to decide whether a proposition of the standard subject-predicate form is true or false, one checks whether the subject is in the extension of the predicate. The proposition is then true if and only if the subject is in the extension of the predicate. The problem is that there is no present king of France, so the present king of France cannot be found on the list of bald things or non-bald things. So, it would appear that the proposition expressed by "The present king of France is bald" is neither true nor false. However, analyzing the relevant concepts and propositions, Russell proposed that what definite descriptions really express are not propositions of the subject-predicate form, but rather they express existentially quantified propositions. Thus, "The present king of France" is analyzed, according to Russell's theory of descriptions, as "There exists an individual who is currently the king of France, there is only one such individual, and that individual is bald." Now one can determine the truth value of the proposition. Indeed, it is false, because it is not the case that there exists a unique individual who is currently the king of France and is bald—since there is no present king of France (Bertolet 1999).

Criticism
While the method of analysis is characteristic of contemporary analytic philosophy, its status continues to be a source of great controversy even among analytic philosophers. Several current criticisms of the analytic method derive from W.V. Quine's famous rejection of the analytic–synthetic distinction. While Quine's critique is well-known, it is highly controversial.

Further, the analytic method seems to rely on some sort of definitional structure of concepts, so that one can give necessary and sufficient conditions for the application of the concept. For example, the concept "bachelor" is often analyzed as having the concepts "unmarried" and "male" as its components. Thus, the definition or analysis of "bachelor" is thought to be an unmarried male. But one might worry that these so-called necessary and sufficient conditions do not apply in every case. Wittgenstein, for instance, argues that language (e.g., the word 'bachelor') is used for various purposes and in an indefinite number of ways. Wittgenstein's famous thesis states that meaning is determined by use. This means that, in each case, the meaning of 'bachelor' is determined by its use in a context. So if it can be shown that the word means different things across different contexts of use, then cases where its meaning cannot be essentially defined as 'unmarried man' seem to constitute counterexamples to this method of analysis. This is just one example of a critique of the analytic method derived from a critique of definitions. There are several other such critiques (Margolis & Laurence 2006). This criticism is often said to have originated primarily with Wittgenstein's Philosophical Investigations.

A third critique of the method of analysis derives primarily from psychological critiques of intuition. A key part of the analytic method involves analyzing concepts via "intuition tests". Philosophers tend to motivate various conceptual analyses by appeal to their intuitions about thought experiments. (See DePaul and Ramsey (1998) for a collection of current essays on the controversy over analysis as it relates to intuition and reflective equilibrium.)

In short, some philosophers feel strongly that the analytic method (especially conceptual analysis) is essential to and defines philosophy—e.g. Jackson (1998), Chalmers (1996), and Bealer (1998). Yet, some philosophers argue that the method of analysis is problematic—e.g. Stich (1998) and Ramsey (1998). Some, however, take the middle ground and argue that while analysis is largely a fruitful method of inquiry, philosophers should not limit themselves to only using the method of analysis.

title: Philosophical analysis


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Talk:Set (mathematics)

Equality
Somebody removed my comment in parenthesis:
 * ...one set equals the other, so the two sets are in fact one and the same...

about what it means for two sets to be equal. Why drop it? Personally, I think it is less than obvious what equality means in broader terms. It would be easy to imagine a world where a different term is used for the same concept in the context of sets, instead of equality, such as congruence.--Jonathan G. G. Lewis 00:28, 20 March 2021 (UTC)

Recent edits seem to have removed some of the less formal language, turning the article into what reads more like a maths textbook. I am not convinced this is an enhancement, from an ordinary, non-mathematician Wikipedia readership perspective. --Jonathan G. G. Lewis 00:28, 20 March 2021 (UTC)

 * Thanks for the comment. It would help other editors understand what you're talking about if you could point to the specific edit that removed this language, so that we'd have the context. I assume it's this edit. The before text reads:
 * Another essential property of sets is that two sets are equal (one set equals the other, so the two sets are in fact one and the same) if and only if every element of each set is an element of the other.
 * and the after text reads:
 * Two sets are equal if and only if every element of each set is an element of the other;
 * I agree with this edit. The old language says the same thing four times:
 * two sets are equal
 * one set equals the other
 * the two sets are in fact one
 * (the two sets are in fact) the same
 * Would you say "two numbers are equal, that is one equals the other, so that they are in fact one number and they are the same number, if ..."
 * On the other hand, I find the wording "every element of each set is an element of the other" a bit confusing and I think we can improve on it. --Macrakis (talk) 00:43, 20 March 2021 (UTC)
 * PS It would also help if you'd sign your comments on this page with your user name, which is apparently Jonazo, not Jonathan G. G. Lewis. --Macrakis (talk) 00:46, 20 March 2021 (UTC)
 * /Jonathan G. G. Lewis: I think I may have been the one to make some of the edits you are talking about. I would be happy to discuss specific edits here, if you like.  Regarding the definition of equality you mention above, for me it was not a matter of formal vs. informal; I was just trying to simplify the wording and eliminate some redundancy, as Macrakis said. Best wishes, Ebony Jackson (talk) 01:43, 20 March 2021 (UTC)
 * Actually, it was other edits I had in mind. It is a question of who is the intended readership. Most non-mathematicians seem to have a mental barrier preventing them digesting maths jargon such as 'if and only if'. Any mathematician should know all this already anyway, so what is the point, unless it is intended to reach a wider readership? As for this specific case of my more verbose explanation of what set equality means, and its removal in favour of a succinct mathematical definition, my view is neutral.--Jonathan G. G. Lewis 01:09, 28 January 2022 (UTC)

Collection
The term "collection" is not defined. Is a collection and a set the same thing? Why does Collection disambiguate to Set? Comfr (talk) 02:00, 23 February 2022 (UTC)
 * A set is the mathematical model of what is called a collection in common language. So, the definition of "collection" can be found in any dictionary. I agree that the first sentence of the article may be confusing, and I'll try to fix it. D.Lazard (talk) 08:48, 23 February 2022 (UTC)

 * Informally "defining" the most basic notion of mathematics is difficult. Halmos ("Naive set theory", German translation, 4th ed., 1976) starts his first chapter like this (my translation back to English): "A herd of wolves, a bunch of grapes, or a swarm of pidgeons are examples for sets of things. The mathematical notion of a set can be seen as the foundation of conteporary mathematics." - Maybe giving a few everyday examples is a good idea: "A set is the mathematical model of a collection of things,[1][2][3] like a herd of wolves, a bunch of grapes, or a swarm of pidgeons.[4=Halmos]"?
 * And maybe, later in the lead, we should mention that axiomatizations of set theory use the term "set" as a basic primitive (like "point" in geometry), that it is not defined, but its properties are fixed by the axioms (like modern geometry does no longer explain what a point is, but rather how it relates to e.g. a line). So those people who are not satisfied with the informal explanation may look at the axioms. - Jochen Burghardt (talk) 17:08, 23 February 2022 (UTC)
 * In practice, "collection" or "family" is typically used for higher-type objects — for example, you might have some sets, and you gather them together in a collection of sets. There's nothing technically wrong with saying "set of sets", but sometimes it's useful to keep track of the type.
 * As an aside, the (first-order) axioms do not actually fix the properties of sets, but only some of their properties. That's why the axiomatic method is insufficient to make the objects of discourse well-specified, whereas it is arguably possible to give a well-specified informal definition.  --Trovatore (talk) 20:24, 24 February 2022 (UTC)

History
Can we move the history section to the end of the article? The history section is somewhat advanced (mentioning classes, Russell's paradox, axioms of set theory), whereas the next few sections on the basic notation and concepts of set theory are more helpful for 99% of readers, I think. Ebony Jackson (talk) 02:31, 8 August 2023 (UTC)
 * Thank you, . By far the worst thing about Wikipedia's coverage of mathematical topics is that many of the articles are written largely by mathematicians who write as though for other mathematicians, often producing results which are incomprehensible to most ordinary readers of the encyclopaedia. It is refreshing to see someone making an attempt to reduce the extent of this problem. 👍 JBW (talk) 10:16, 4 September 2023 (UTC)
 * Yes, moving the history section to the end of the article, as you have now done, is a big improvement. Paul August ☎ 12:46, 4 September 2023 (UTC)

The lead of this article
Some months ago I added a "set of cows" as an example to the lead, and mentioned sets of sheep in my edit summary. It was reverted by with the comment "You never saw a mathematical set of sheaps, you saw a group of sheaps tha could modeled by a mathematical set (but not by a mathematicl group)". The comment above by quoting Halmos, leads me to revisit this. I think that the notion that collections of non-mathematical objects cannot be sets but can only be modeled by sets is (1) wrong, (2) not supported by sources and contradicted by many eminent authorities such as Halmos, (3) an obstacle to the understanding of this basic concept, since it is sets of everyday objects that will be most easily understood by mathematically unsophisticated readers, (4) a source of absurdity (try to actually model a collection of cows by a set without using the notion of bijection, which is only defined between sets). So I propose to remove the assertion that the elements of sets can only be "mathematical objects". McKay (talk) 05:06, 14 March 2024 (UTC)

 * This article is entitled "Set (mathematics)"; so, it is about mathematical sets, not about the English meaning of the word. More precisely, the mathematical concept of a set is the abstraction of the usual concept of collection. However, intuitive examples are fundamental for understanding the concept of a set. Similarly, a line drawn on a paper sheet is not a mathematical line, although the first gives an intuition of the second, and the second is an abstraction of the first (and other examples). Confusing the physical reality with its mathematical abstraction is error prone. For example, "a set is larger than a subset" is true in everyday world, but not in the mathematical world, as soon as infinite sets are considered.
 * Said otherwise, if a set contains a non-mathematical object, this is not a mathematical set. D.Lazard (talk) 10:48, 14 March 2024 (UTC)

 * The reason for the title is to distinguish the article from other uses of the word "set", of which there are very many (see the disambiguation page set). It is not to remove from consideration some of the things that are sets. Your example doesn't work: a finite set of everyday objects behaves just the same as a finite set of mathematical objects and there is no reason to draw a line between them. Your distinction between "set" and "mathematical set" is not made by any authority that I know of. Infinite sets are a more advanced topic that is for us to explain, but there are infinite sets of everyday objects too, such as the set of all possible paragraphs. I'm waiting for others to comment, but at the moment I don't believe you have a case. McKay (talk) 00:12, 15 March 2024 (UTC)
 * I did pings wrong before, here is a repeat attempt: . McKay (talk) 00:16, 15 March 2024 (UTC)
 * I seem to recall that at some point Penelope Maddy proposed an ontology whereby there were no pure sets at all; sets could contain real-world objects and other sets, but at some point the sets had to bottom out into urelements (which were, I think, supposed to be physical objects? I'm a little unsure on that point). Her work is in philosophy of mathematics, so these were definitely supposed to be mathematical sets.
 * In any case I think it's at least controversial to say that a set in the sense of mathematics is restricted to containing only mathematical objects. --Trovatore (talk) 01:05, 15 March 2024 (UTC)
 * I don't have a firm opinion yet, but here a some thoughts.
 * Halmos says (on the same page as my above citation) "An element of a set can be a wolf, a grape, or a pidgeon" - however, he says this on the very first page of his introduction, maybe just as an informal motivation. If I understood correctly, he'd have no problem using such analogies ("a set is like a herd of wolves...") in a motivation, before turning to strictly formal definitions.
 * Concerning the latter, I looked at the table of axioms given by Halmos at the end of his book, and they leave the question open what a set can be and what an element can be. It seems to me that the minimal model of these axioms contains only mathematical objects, more precisely: objects that can be built from the empty set (existing as a consequence of ax.2) and the infinite set (required by ax.6). However, other models may well include elements that aren't sets. Whether or not e.g. a real wolf (or the notion of it, or the reference to it, or the name of it, or whatever of it) can be such an element, seems to be a philosophical question. I feel that it can be convenient to allow real-world things as set elements, e.g. in Russell's analysis of the sentence "The present King of France is bald"; while he actually used predicate logic, one can imagine a corresponding set-theoretic argument. - Jochen Burghardt (talk) 18:57, 16 March 2024 (UTC)

More griping about the lead
While I'm at it, the lead has a more serious problem. Try to imagine an average high school student attempting to understand this. We start with "A set is the mathematical model for a collection of different things." Clicking on mathematical model, we read "A mathematical model is an abstract description of a concrete system using mathematical concepts and language." WTF? At this point our average student gives up, and yet the concept of "set" is one of the simplest to explain and an average primary school student can understand it. Now read it again carefully: this sentence says that a set of integers is not actually a collection of integers but a mathematical model of a collection of integers. Ridiculous! What the lead should start with is "A set is the mathematical concept corresponding to a collection of distinct things". Then it can continue with examples, including examples of sets of everyday objects. McKay (talk) 01:32, 15 March 2024 (UTC)

 * I agree to omit "mathematical model for". The article mathematical model seems to apply to e.g. sets of differential equations modelling climate, and is not too appropriate here. As for "things", this is discussed in section . - Jochen Burghardt (talk) 18:01, 16 March 2024 (UTC)
 * I agree that, here "mathematical model" is pure pedantry for saying that, in mathematics, the concept of a set is a mathematical abstraction of the concept of a collection. Even the latter formulation must be avoided here, since the concept of abstraction is philosophy, not mathematics. So, I have changed the first sentence into
 * The addition of the word "typically" allows avoiding the philosophical question of whether one can talk of a set of cows. I have revrted also the order, in the first paragraph, between infinite sets, singletons and teh empty set (set theory would not exist without infinite sets). D.Lazard (talk) 16:38, 18 March 2024 (UTC)
 * The addition of the word "typically" allows avoiding the philosophical question of whether one can talk of a set of cows. I have revrted also the order, in the first paragraph, between infinite sets, singletons and teh empty set (set theory would not exist without infinite sets). D.Lazard (talk) 16:38, 18 March 2024 (UTC)

Non-distinct elements
I was once chastised for thinking a set could contain duplicated elements, e.g. something like {2,2,3}. Those elements are the prime factorization of 12, and if it makes sense for them to be a set, that would be useful. E.g., the least common denominator of two integers is simply (the product of the elements of) the intersection of their prime factorization "sets", and the product of two integers is (the product of the elements of) the union of their prime factorization "sets".

Can that be salvaged in any way?

BMJ-pdx (talk) 03:46, 28 April 2024 (UTC)

 * See Multiset. D.Lazard (talk) 07:49, 28 April 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Set (mathematics)


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Internet backbone

The Internet backbone is the principal data routes between large, strategically interconnected computer networks and core routers of the Internet. These data routes are hosted by commercial, government, academic and other high-capacity network centers as well as the Internet exchange points and network access points, which exchange Internet traffic internationally. Internet service providers (ISPs) participate in Internet backbone traffic through privately negotiated interconnection agreements, primarily governed by the principle of settlement-free peering.

The Internet, and consequently its backbone networks, do not rely on central control or coordinating facilities, nor do they implement any global network policies. The resilience of the Internet results from its principal architectural features, such as the idea of placing as few network state and control functions as possible in the network elements, instead relying on the endpoints of communication to handle most of the processing to ensure data integrity, reliability, and authentication. In addition, the high degree of redundancy of today's network links and sophisticated real-time routing protocols provide alternate paths of communications for load balancing and congestion avoidance.

The largest providers, known as Tier 1 networks, have such comprehensive networks that they do not purchase transit agreements from other providers.

Infrastructure

The Internet backbone consists of many networks owned by numerous companies.

Fiber-optic communication remains the medium of choice for Internet backbone providers for several reasons. Fiber-optics allow for fast data speeds and large bandwidth, suffer relatively little attenuation — allowing them to cover long distances with few repeaters — and are immune to crosstalk and other forms of electromagnetic interference.

The real-time routing protocols and redundancy built into the backbone is also able to reroute traffic in case of a failure. The data rates of backbone lines have increased over time. In 1998, all of the United States' backbone networks had utilized the slowest data rate of 45 Mbit/s. However, technological improvements allowed for 41 percent of backbones to have data rates of 2,488 Mbit/s or faster by the mid 2000s.

History
The first packet-switched computer networks, the NPL network and the ARPANET were interconnected in 1973 via University College London. The ARPANET used a backbone of routers called Interface Message Processors. Other packet-switched computer networks proliferated starting in the 1970s, eventually adopting TCP/IP protocols or being replaced by newer networks.

The National Science Foundation created the National Science Foundation Network (NSFNET) in 1986 by funding six networking sites using 56 kbit/s interconnecting links, with peering to the ARPANET. In 1987, this new network was upgraded to 1.5 Mbit/s T1 links for thirteen sites. These sites included regional networks that in turn connected over 170 other networks. IBM, MCI and Merit upgraded the backbone to 45 Mbit/s bandwidth (T3) in 1991. The combination of the ARPANET and NSFNET became known as the Internet. Within a few years, the dominance of the NSFNet backbone led to the decommissioning of the redundant ARPANET infrastructure in 1990.

In the early days of the Internet, backbone providers exchanged their traffic at government-sponsored network access points (NAPs), until the government privatized the Internet and transferred the NAPs to commercial providers.

Modern backbone
Because of the overlap and synergy between long-distance telephone networks and backbone networks, the largest long-distance voice carriers such as AT&T Inc., Verizon, Sprint, and Lumen also own some of the largest Internet backbone networks. These backbone providers sell their services to Internet service providers.

Each ISP has its own contingency network and is equipped with an outsourced backup. These networks are intertwined and crisscrossed to create a redundant network. Many companies operate their own backbones which are all interconnected at various Internet exchange points around the world. In order for data to navigate this web, it is necessary to have backbone routers—routers powerful enough to handle information—on the Internet backbone that are capable of directing data to other routers in order to send it to its final destination. Without them, information would be lost.

Peering agreements
Backbone providers of roughly equivalent market share regularly create agreements called peering agreements, which allow the use of another's network to hand off traffic where it is ultimately delivered. Usually they do not charge each other for this, as the companies get revenue from their customers.

Regulation
Antitrust authorities have acted to ensure that no provider grows large enough to dominate the backbone market. In the United States, the Federal Communications Commission has decided not to monitor the competitive aspects of the Internet backbone interconnection relationships as long as the market continues to function well.

Transit agreements
Backbone providers of unequal market share usually create agreements called transit agreements, and usually contain some type of monetary agreement.

Egypt
During the 2011 Egyptian revolution, the government of Egypt shut down the four major ISPs on January 27, 2011 at approximately 5:20 p.m. EST. The networks had not been physically interrupted, as the Internet transit traffic through Egypt was unaffected. Instead, the government shut down the Border Gateway Protocol (BGP) sessions announcing local routes. BGP is responsible for routing traffic between ISPs.

Only one of Egypt's ISPs was allowed to continue operations. The ISP Noor Group provided connectivity only to Egypt's stock exchange as well as some government ministries. Other ISPs started to offer free dial-up Internet access in other countries.

Europe
Europe is a major contributor to the growth of the international backbone as well as a contributor to the growth of Internet bandwidth. In 2003, Europe was credited with 82 percent of the world's international cross-border bandwidth. The company Level 3 Communications began to launch a line of dedicated Internet access and virtual private network services in 2011, giving large companies direct access to the tier 3 backbone. Connecting companies directly to the backbone will provide enterprises faster Internet service which meets a large market demand.

Caucasus
Certain countries around the Caucasus have very simple backbone networks. In 2011, a 70-year-old woman in Georgia pierced a fiber backbone line with a shovel and left the neighboring country of Armenia without Internet access for 12 hours. The country has since made major developments to the fiber backbone infrastructure, but progress is slow due to lack of government funding.

Japan
Japan's internet backbone requires a high degree of efficiency to support high demand for the Internet and technology in general. Japan had over 86 million Internet users in 2009, and was projected to climb to nearly 91 million Internet users by 2015. Since Japan has a demand for fiber to the home, Japan is looking into tapping a fiber-optic backbone line of Nippon Telegraph and Telephone (NTT), a domestic backbone carrier, in order to deliver this service at cheaper prices.

China
In some instances, the companies that own certain sections of the Internet backbone's physical infrastructure depend on competition in order to keep the Internet market profitable. This can be seen most prominently in China. Since China Telecom and China Unicom have acted as the sole Internet service providers to China for some time, smaller companies cannot compete with them in negotiating the interconnection settlement prices that keep the Internet market profitable in China. This imposition of discriminatory pricing by the large companies then results in market inefficiencies and stagnation, and ultimately affects the efficiency of the Internet backbone networks that service the nation.

title: Internet backbone


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Talk:Zipf's law

Needs simple leading statement
The article needs to start with a very simple leading statement, so that anyone stumbling onto it can know what it is about. The discussion can then move onto more technical stuff.
 * Provided one. --Jorge Stolfi (talk) 01:24, 9 May 2023 (UTC)

True that "the" is twice as common as the next one?
Is it true that the word "the" does indeed occur about twice as often as the next common English word? The rest of the article seems to allow for some proportionality constants. AxelBoldt

 * It's not true, so I replaced it with a statement about Shakespeare's plays. AxelBoldt
 * It is approximately true, see the example of the Brown corpus. The normalization constant applies to both words, so the ratio is 1/1 to 1/2, that is 2:1. --Jorge Stolfi (talk) 01:28, 9 May 2023 (UTC)

Random typing
The main article claimed that I doubt that very much. For one thing, if you type randomly, all words of length one will be equally likely, all words of length 2 will be equally likely and so on. Or am I missing something? Maybe we should perform a little perl experiment. AxelBoldt
 * the frequency distribution of words generated by random typing follows Zipf's law.

 * I tried this Python code, and plotted the results in a log-log plot. The early ranks are a bit stepped, but the overall pattern fits Zipf's law rather well. The Anome

 * Could you repeat the experiment with all letters and the space getting the same probability? That's at least what I thought off when I heard "random typing". AxelBoldt

 * When I was told of the "random typing" experiment, the "typing" part was more important than "random". If you sit at a keyboard and randomly type, you have a much higher chance of hitting certain keys than others because your fingers tend to like certain positions.  Also, the human brain is a pattern-matching machine, so it works in patters.  If you look at what you type, you might notice you tend to type the same sequences of letters over and over and over.
 * For a brief historical note, this theory was used in cracking the one-time-pad cyphers. Humans typed the pads, so it was possible to guess at the probability of future cyphers by knowing past ones.  It is like playing the lottery knowing that a 6 has a 80% chance of being the first number while a 2 has a 5% chance.  If you knew the percentage chance for each number and each position, you have a much greater chance of winning over time. Kainaw 13:59, 24 Sep 2004 (UTC)

I can't right now, but I'll give you the reason for the skewed probabilities -- the space is by far the most common character in English, and other chars have different probabilities -- I wanted to model that. The Anome

The main article claimed that ''I doubt that very much. For one thing, if you type randomly, all words of length one will be equally likely, all words of length 2 will be equally likely and so on. Or am I missing something?''
 * the frequency distribution of words generated by random typing follows Zipf's law.

Yes, you're missing something.

It doesn't *exactly* match Zipf's law. But then, no real measurement exactly matches Zipf's law -- there's always "measurement noise". It does come pretty close. In most English text, about 1/5 of all the characters are space characters. If we randomly type the space and 4 other letters (with equal letter frequencies), then we expect words to have one of these discrete probabilities: 1/4*1/5: each of the 4 single-letter words 1/4*1/4*1/5: each of the 4*4 two-letter words 1/4*1/4*1/4*1/5: each of the 4*4*4 three-letter words. ... (1/5)*(1/4)^n: each of the 4^n n-letter words.

This is a stair-step graph, as you pointed out. However, if we plot it on a log-log graph, we get x = log( 4^n / 2 ) = n * log(4) - log(2). y = log( 1/5 * (1/4)^n ) = -n*log(4) - log(5).

which is pretty close to a straight line (and therefore a Zipf distribution), with slope m = Δy / Δx = -1.

You get the same stair-step on top of a straight line no matter how many letters (plus space) you use, with equal letter frequencies. If the letter frequencies are unequal (but still memoryless), I think that rounds off the corners and makes things even closer to a Zipf distribution. --DavidCary 01:54, 12 Feb 2005 (UTC)

The problem here seems to be arising from the ambiguarity of the phrase "random typing".To avoid this I'll replace the phrase with "random sampling of a text". Jekyll

It's already gone, never mind. Jekyll 14:45, 18 November 2005 (UTC)

 * this is a pretty minor example of Zipf's law. A much more important one is that the frequency of occurrence of words in natural language conform to Zipf's law. This is in the original 1949 reference. Callivert (talk) 11:29, 24 March 2008 (UTC)

I did some study for university on Zipf's Law. A (in my opinion) very clever and better explanation for Zipf's law is presented by Baek et al. I do not have the time to add this information. In case anyone can do this: http://stacks.iop.org/1367-2630/13/i=4/a=043004 and https://doi.org/10.1038/474164a — Preceding unsigned comment added by 2A02:810D:4740:5520:E1D3:6110:57E:C5B7 (talk) 16:14, 11 December 2017 (UTC)

Why ?
Moved from main article:

 * We need an explanation here: why do these distributions follow Zipf's law?

 * No we don't. Zipf's law is empirical, not theoretical.  We don't know why it works.  But even without a theory, even the simplest experiments that try to model a society of independent actors consistently turn it up!--BJT

Well, empirical facts have to be explained too. It's not enough to simply state that the moon always shows us the same side; you have to give the reason if you try to understand the world. It's the same here. If Zipfian distributions show up in a variety of situations, then there must be some underlying principle which generates them. I doubt very much that "we don't know" that principle. AxelBoldt

 * I agree - every theory begins with empirical evidence. The theory models an explanation to fit those facts. I'm sure someone has tried to come up with an explanation? 70.93.249.46

 * We need an explanation here: why do these distributions follow Zipf's law?

An excellent question. However, I doubt there is a single cause that can explain every occurance of Zipf's law. (For some distributions, such as wealth distribution, the cause of the distribution is controversial).

''Well, empirical facts have to be explained too. It's not enough to simply state that the moon always shows us the same side; you have to give the reason if you try to understand the world.''

Good point. However, sometimes we don't yet know the cause of some empirical facts -- we can't yet give a good explanation. In those cases, I would prefer the Wikipedia article to bluntly tell me "we don't know yet" rather than try to dance around that fact.

While it is true that Zipf's law is empirical, I agree with AxelBoldt that it is useful to have an interpretation of it. The most obvious place to look is the book that Zipf himself wrote in which he linked his observation to the Principle of Least-Effort, kind of an application of Conservation of Energy to human behavior.

I've just measured Polish Wikipedia page access distribution using Apache logs (so they had to be heavily Perlscripted) for about 2 weeks of late July, only for main namespace articles, and excluding Main Page. In most part it seems to be following Zipf's law with b about 0.5, except at both ends, where it behaves a bit weird (what was to be expected). Now why did I get constant so grossly different from stated constant for English Wikipedia ?

Some possibilities: Taw 01:38, 4 Aug 2003 (UTC)
 * Polish and English Wikipedias really have different Zipf's factors
 * It was due to my perlscripting
 * Measurment given for English Wikipedia here is wrong for some reason, like measuring only top 100, and not all articles.

I think we may be fast approaching the point when merging this article with Zipf-Mandelbrot law would be appropriating, along the way doing some reorganizing of the article. Michael Hardy 22:57, 6 Dec 2003 (UTC)

Although the reason is not well-understood, mechanisms that bring about the Zipf distribution have been suggested by physicists. Power laws tend to crop up in systems where the entities (words in this case) are not independent, but interact locally. The choice of a word isn't random, neither does it follow a mechanistic prescription - the choice of a word depends strongly on what other words have already been chosen in the same sentence/paragraph. I think these speculations should be mentioned in the article as a side note, for the sake of completeness. 137.222.40.132 12:45, 17 October 2005 (UTC)

It is meaningful to ask why in general a particular distribution is found in nature - the passage above is a good start; I'd like more clarification - for example, the normal distribution arises when their the outcome is caused by a large number of minor factors where no particular factor predominates. The bimodal distribution arises when their are a large number of minor factors coupled with one predominant factor. The Poisson distribution arises when an event is the consequence of a large number of rare events converging. Etc. For Zipf's distribution, I would like to know, why does interdependence of events lead to it?

Another interesting example of Zipf's law is the distribution of 25,000,000 organic compounds among the possible shapes that the ring structures can take (download PDF or HTML). Half of all the compounds have just 143 shapes (out of more nearly 850,000 shapes). The authors refer to a "rich-get-richer" reason: When we are speaking or writing, the probability that we will use a particular word is proportional to the number of times that we have already used it. The particular reason for organic chemistry is that is quicker and cheaper to synthesize new compounds if you can buy the "parts" commercially, or if syntheses for the "parts" have already been published; thus the most common shapes tend to be used more.

In these word-list examples we also have to do with a fundamental fact of linguistics: Every language has a few dozen "structural" words, mostly articles, conjunctions, and prepositions, which have to be used to achieve proper syntax (note the three most popular words in English). This can skew the distribution of the most popular words.

This would also apply to populations of cities. A large city has many opportunities for work and leisure, so it will attract new inhabitants and keep its current ones. On the whole, one would expect that the probability that new inhabitants will come would be, very roughly, proportional to the existing population.

So the reason seems to be that prior use encourages further use. Almost banal.

--Solo Owl (talk) 17:26, 7 September 2008 (UTC)

Consistency of variables in text
In the examples section, the variable quoted in each case is b. However, this variable is not used anywhere else. Some consistency throughout the article would be nice (and less confusing!). — 130.209.6.41 17:01, 1 Jun 2004 (UTC)

 * I agree - please explain what is b ? \Mikez 10:00, 8 Jun 2004 (UTC)

 * I was about to add something about this as well... is it what is called s in the discussion of formulas? It's not clear from the text. -- pne 14:01, 8 Jun 2004 (UTC)

 * Well, Ive seen Zipf's Law stated as $$f_n=[\mbox{constant}]/n^b.$$ So Im pretty sure that s and b are the same thing. Im changing b to s in the "Examples..." section. -- Aparajit 06:01, Jun 24, 2004 (UTC)

 * Can someone check the values of b / s given in the examples? Especially the word frequency example.  I took the data for word frequencies in Hamlet, and fitted a line to the log-log plot.  This gave a slope of more like 1.1, rather than the 0.5 figure quoted here.  Taking the merged frequencies over the complete set of plays gives a value which gets towards 1.3.  This would agree more with the origin of Zipf's law, which is that the frequency of the i'th word in a written text is proportional to 1/i.  The value of 0.5 seems much too small to match this observation.  Graham.

Linked site doesn't exist
It seems to me that the special page of the [|most popular pages] no longer exists, but it's used for an example in this page.

Has this page simply moved or do we need to get a new example?

reported constants in Examples section
the examples section reports values of s < 1 as resulting from analysis of wikipedia page view data. the earlier discussion correctly notes that such values do not yield a valid probability distribution. what gives? perhaps (s - 1) is being reported?

 * Or it could be that that value of s is right for a moderately large (hundreds?) finite number of pages. That seems to happen with some usenet posting statistics. Michael Hardy 22:07, 28 Jun 2004 (UTC)

I miss reference to Zipf's (other) law: the principle of least effort.

s = 1?
I plotted Shakespeare's word frequency lists and top 5000 words in Wikipedia: Where did the old value of s~0.5 come from? -- Nichtich 00:27, 23 Jun 2005 (UTC)

Sources needed for examples
Section 3 ("Examples of collections approximately obeying Zipf's law") has a bunch of examples with no further explanation or reference (Shakespeare excluded). That's highly undesirable, so let's get some sources. By the way, I find the final point (notes in a musical performance) very questionable. I imagine it would depend very much on the type of music. EldKatt (Talk) 13:20, 19 July 2005 (UTC)

 * It might not be a bad idea to give examples from Zipf's book. Also see the article by Richard Perline in the February 2005 issue of Statistical Science. Michael Hardy 22:28, 19 July 2005 (UTC)

Too technical?
Rd232 added the "technical" template to the article. I've moved it here per template. Paul August ☎ 03:49, 27 November 2005 (UTC)

 * There have been numerous edits to the article since the template was first added almost a year ago. Also, there has been no discussion of what about the article is too technical so I've removed the template.  Feel free to put it back, but if you do, please leave some comments as to what you find is too technical and some suggestions as to how to improve the article.  Lunch 02:25, 21 November 2006 (UTC)

Does Wikipedia traffic obey Zipf's law?
Yes, apparently, with an exponent of 0.5. See Does Wikipedia traffic obey Zipf's law? for more. -- The Anome 22:45, 20 September 2006 (UTC)

 * Why 0.5? And what else has the exponent? I've got 0.5 in some genomic dta, don't know why its 0.5 and not 1.0 .... 67.198.37.16 (talk) 18:17, 24 February 2020 (UTC)

Wikipedia's Zipf law
Just a plot of English Wikipedia word frequencies: http://oc-co.org/?p=79

 * Is this plot availiable to Wikipedia - i.e. Free content? It would look good in the article.
 * Yes, it is released under LGPL by me, the author :) --  Victor Grishchenko

 * I downloaded it, tagged it as LGPL with you as author, and put it into the article - please check it out and make corrections if needed. This is an excellent demonstration of Zipf's law (and its limitations). Thanks! PAR 15:00, 29 November 2006 (UTC)

--

This article has 107 "the" and 68 "of" last time i checked.

-- —Preceding unsigned comment added by 200.198.220.131 (talk) 13:35, 5 October 2009 (UTC) Now this is interesting. It is, however, not a true power law, as Grishchenko shows quite clearly by superposing blue and green lines. That is, English Wikipedia is not an example of Zipf’s law. The tail (the lower righthand part of the curve) is not typical of power laws.

The first part of the plot, for the 8000 or so most common words, does follow a power law, with exponent slightly greater than 1, just as we would expect from Zipf’s Law. The rest of the plot, for another million different words in English Wikipedia, follows a power law with exponent approximately 2; this part has the tail that we look for in power laws.

My guess is that in writing encyclopedia articles, one must select from the 8000 most common words just to create readable prose. Hence the log-log plot with slope ~1.

An encyclopedia, by its nature, is also going to contain hundreds of thousands of technical terms and proper names, which will be used infrequently in the encyclopedia as a whole. Hence the log-log plot with a steeper slope. Why should this slope be ~2? Does anyone know?

Does Zipf’s law apply to individual Wikipedia articles? Do other encyclopedias behave the same?

--Solo Owl (talk) 17:59, 7 September 2008 (UTC)
 * It's a general principle that words drop off more quickly than Zipf's law predicts. If we looked at word n-grams I think we would see something closer. Dcoetzee 21:27, 31 October 2008 (UTC)

+ 1 paper related to Wikipedia's Zipf law:
 * Index wiki database: design and experiments. Krizhanovsky A. A. In: FLINS'08, Corpus Linguistics'08, AIS/CAD'08, 2008. -- AKA MBG (talk) 17:52, 3 November 2008 (UTC)

On "Zipf, Power-laws, and Pareto - a ranking tutorial"
I've found two doubtful places in the tutorial by L.Adamic (ext. link N3). Probably, I've misread something...

First, "(a = 1.17)" regarding to Fig.1b must be a typo; the slope is clearly -2 or so.

Second, it is not clear whether Fig.2a is a cumulative or disjoint histogram? To the best of my knowledge, logarithmically-disjointly-that-way-binned Zipf must have slope=-1 and not -2. I.e., if every bin catches items which popularity resides in the range $$[c^{i}:c^{i+1})$$. Just to verify it, I did a log2-log2 graph of log2-binned word frequencies compiled from Wikipedia. I.e. y is log2 of number of words mentioned $$2^x$$ to $$2^{x+1}-1$$ times in the whole Wikipedia. Although the curve is not that simple, it shows slope=-1, especially in regard to more frequent words.

Any thoughts? Gritzko

 * Yes it looks like a typo, a=1.17 certainly is not right. Regarding Fig 2a, it is cumulative (the vertical axis is "proportion of sites" which will have an itercept at (1,1)). The Zipf law does not specify an exponent of -1, just that it is some negative constant. It happens to be close to -1 for word frequency, but maybe its closer to -2 for aol user data. PAR 14:20, 1 January 2007 (UTC)

 * "As demonstrated with the AOL data, in the case b = 1, the power-law exponent a = 2.", i.e. b is "close to unity" in the case of AOL user data. I had some doubts whether Fig 2a is cumulative because at x=1 y seems to be slightly less than 1. Probably, it is just a rendering glitch. Thanks! -- Gritzko

k = 0 in support?
I'm not sure if k=0 should be included in the support... pmf is not well defined there, as 1 / (k^s) = 1 / 0 and it -> +inf. Krzkrz 08:56, 3 May 2007 (UTC)

biographical information
I was surprised that there wasn't even a brief note at the beginning of the article on who Zipf is (was?). I think it's sort of nice to see that before you get into the technical stuff. Jdrice8 05:38, 14 October 2007 (UTC)

 * That was the brief second paragraph of the article. Now I've made it into a parenthesis in the first sentence, set off by commas. Michael Hardy 01:58, 15 October 2007 (UTC)

Word length
I'm sure I've come across in a no. of places the term Zipf's law used to refer to the inverse relation between word length & frequency. This may well be a misuse, but isn't it common enough to be mentioned with a link to whatever the correct term is? Peter jackson (talk) 14:43, 15 September 2008 (UTC)

Same here. I wish the page explained that up front, but I don't know enough to write it. — Preceding unsigned comment added by 76.94.255.117 (talk) 20:48, 14 July 2012 (UTC)

This page supposedly has a quote from Zipf: http://link.springer.com/content/pdf/10.1007%2F978-1-4020-4068-9_13.pdf

“that the magnitude of words tends, on the whole, to stand in an inverse (not necessarily proportionate) relationship to the number of occurrences” — Preceding unsigned comment added by 216.239.45.72 (talk) 19:43, 16 May 2013 (UTC)

The word-length sense of the term "Zipf's law" is commonly recognized in the psycholinguists and HCI communities. It would certainly not be difficult to find this usage in the literature of those fields. I don't see how this could be considered a "misuse". It should be handled as a separate sense of the term.

Lead
The lead must be rephrased: defining Zipf's law in terms of a Zipfian distribution is no help to the reader who must actually learn. Srnec (talk) 15:11, 31 October 2008 (UTC) —Preceding unsigned comment added by 81.185.244.190 (talk) 21:28, 8 January 2009 (UTC)

 * I was just coming by to say the same thing. The law itself should be defined in the lede; defining it in terms of something derived from it does no one any good. It's been three and a half years and no one's done anything about it? 174.57.203.45 (talk) 19:12, 1 June 2012 (UTC)

 * Fixed now, I hope. --Jorge Stolfi (talk) 01:32, 9 May 2023 (UTC)

Nonsense? Err Nevermind
The article currently states the following:

 * That Zipfian distributions arise in randomly-generated texts with no linguistic structure suggests that in linguistic contexts, the law may be a statistical artifact.[2]

I don't get it, this sounds like patent non-sense. When one generates a random text, one must choose works with some random frequency or distribution. What distribution is used? The result of random generation should show the same distribution as the random number generator: so if one generates random text using a Zipfian distribution, the result will be Zipfian. So this is a tautology. Surely something different is being implied in citation 2. However, this text, as written, is nonsense. linas (talk) 22:36, 14 March 2009 (UTC)

 * Ah. The one sentence summary was misleading. I modified the article text so that is made sense. The "random text" is actually a random string, chosen from an alphabet of N letters, with one of the letters being a blank space. The blank space acts as a word separator, and the letter frequencies are uniformly distributed. linas (talk) 22:59, 14 March 2009 (UTC)
 * As the text is now, it's still (or again?) nonsense. It claims that the Zipfian distribution is a consequence of using letters to spell words. Just a few sentences above, the article says that words in the corpus are in Zipfian distribution. This has nothing to do with how words are encoded, and the distribution would exist regardless of what, if anything, is used to represent them. 193.77.151.137 (talk) 16:07, 5 April 2009 (UTC)

Frequency of words in English
The article originally stated,

In English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to $$1/{n^s}$$ where s is just slightly more than one.[citation needed]

This is misleading, not just because the citation is missing, but also because it seems to me that English word frequencies may well follow the maximum entropy principle: their distribution maximises the entropy under the constraint that the probabilities decrease. Zipfs law, for any $$s>1$$, does not maximise the entropy, because it is possible to define distributions with probabilities that decrease even more slowly. For example, $$P(n) = 1/\log(n+1) - 1/\log(n+2)$$ is a very slow one.

I therefore rewrote this bit somehwat. —Preceding unsigned comment added by 131.111.20.201 (talk) 09:52, 1 May 2009 (UTC)

Problems
I think there is an error in the example log-log graph of Wikipedia word frequency vs. rank. The two different lines of color cyan and blue cannot both represent 1/x^2, as they are labelled. I suspect that one is 1/x^3. Whoever has access to the code that generated the graph, please fix it!!!

—Preceding unsigned comment added by 98.222.58.129 (talk) 04:45, 8 January 2010 (UTC)

 * I agree the cyan plot is wrong but I don't think it can be 1/x3 since it nas the same slope as 1/x2. It is more like 10110/x2.  I am going to ping the creator on this - it definitely needs fixing or removing.  Spinning  Spark  17:01, 4 June 2011 (UTC)

 * Actually, all the guide plots are up the creek: they should all be going through the point (1,1). I suspect the author means $$\mathcal{O} f(x)$$ rather than $$ \sim f(x)$$  Spinning  Spark  17:22, 4 June 2011 (UTC)

The section entitled "Statistical Explanation" is clearly nonsense. Equal probability for selecting each letter means that the top of the rank order will be 26 equally probable single letter words. "b" will not occur twice as often as "a". What is Wentian Li talking about??? Clearly it is not what is said here.

—Preceding unsigned comment added by 98.222.58.129 (talk) 04:56, 8 January 2010 (UTC)

citation needed
I don't know how to add a citation needed link. It's needed for the parenthetical, (although the law holds much more strongly to natural language than to random texts)

sbump (talk) 21:42, 19 March 2010 (UTC)

 * type citation needed, it will look like this: . Cheers, — sligocki (talk) 03:21, 21 March 2010 (UTC)

Redundancy
Although the Shannon-Weaver entropy is listed in the table here, I think it is of fundamentally more importance and instructional value to list the REDUNDANCY which, in Zipf's law (s=1) is CONSTANT (and slightly less than 3). Credit to Heinz von Foerster for deriving this from fundamental principles.

98.222.58.129 (talk) 14:04, 19 February 2010 (UTC)

Dr. Ioannides's comment on this article
Dr. Ioannides has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"Entry ignores the massive literature in urban economics, which has received enormous popularity especially after Paul Krugman drew renewed attention to it, namely that city size are Pareto, and often close Zipf distributed, in many countries and historical periods. Thus, I think it would be very important and more useful if the entry made references to some representative studies. Here are two: Gabaix, Xavier. "Zipf's Law for Cities: An Explanation." The Quarterly Journal of Economics, Vol. 114, No. 3 (Aug., 1999). And, more recently, Rossi-Hansberg, Esteban, and Mark L. J. Wright. Urban Structure and Growth April 2007, Review of Economic Studies, 74:2, 597-624 and  Ioannides, Yannis M., and  Spyros Skouras. “US City Size Distribution: Robustly Pareto, but Only in the Tail.” Journal of Urban Economics,  73, No. 1 (January,  2013) 18–29."

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Ioannides has published scholarly research which seems to be relevant to this Wikipedia article:

 * Reference : Yannis M. Ioannides & Spyros Skouras, 2009. "Gibrat's Law for (All) Cities: A Rejoinder," Discussion Papers Series, Department of Economics, Tufts University 0740, Department of Economics, Tufts University.

ExpertIdeasBot (talk) 04:53, 16 June 2016 (UTC)

 * The article Power law seems like a better fit for everything that is described by a power law. 67.198.37.16 (talk) 18:22, 24 February 2020 (UTC)

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Zipf distribution or Zipfian distribution?
Both the phrases "Zipf distribution" and "Zipfian distribution" seem interchangeable but which is preferred?

80.194.75.37 (talk) 09:39, 27 January 2017 (UTC)

Should it say signal, not symbol?
The bit about information theory in the Applications section seems like it makes more sense if the word 'symbol' is changed to 'signal.' I don't know anything about the topic, though so I don't want to change it myself.
 * Word 'symbol' is generally used in information theory - I have added '(event, signal)'. 188.146.143.213 (talk) 09:26, 27 April 2017 (UTC)

 * The section was removed pending clarification and sourcing; see below. --Jorge Stolfi (talk) 01:33, 9 May 2023 (UTC)

Relation to the Zeta distribution
Several times in both this article and Zeta distribution is it stated that while they appear similar and are often referred to by the same name, they aren't related distributions. However, given the probability mass distributions of both distributions, I don't see why they aren't related.

The pmf of Zipf(s,N) is $$f_{Zipf}(k;s,N)=\frac{1/k^s}{\sum\limits_{n=1}^N (1/n^s)}$$, while the pmf of Zeta(s) is $$ f_{\zeta}(k;s) = \frac {1/l^s}{\zeta(s)} $$.

Now, the Riemann zeta function has a very simple form when $$ s > 1 $$, being $$ \zeta(s)= \sum\limits_{n=1}^\infty (1/n^s) $$ (from Riemann zeta function).

This tells me, at least, that the pmf of Zeta(s) is simply the pmf of of Zipf(s,N) in the limit of infinite elements, $$ \lim_{N\to\infty} f_{Zipf}(k;s,N) = f_{\zeta} $$.

Therefore, I am wondering if anyone has a source or (sketch of) proof that shows they aren't related in this manner, so we can cite it - or correct the article to imply that they are in fact directly related through the aforementioned limit. EpicScizor (talk) 13:56, 13 March 2021 (UTC)

Removed unsourced section "Information theory"
The following unsourced and unclear section was deleted:
 * Information theory
 * In information theory, a symbol (event, signal) of probability $$p$$ contains $$-\log_2(p)$$ bits of information. Hence, Zipf's law for natural numbers: $$\Pr(x) \approx 1/x$$ is equivalent with number $$x$$ containing $$\log_2(x)$$ bits of information. To add information from a symbol of probability $$p$$ into information already stored in a natural number $$x$$, we should go to $$x'$$ such that $$\log_2(x') \approx \log_2(x) + \log_2(1/p)$$, or equivalently $$x' \approx x/p$$. For instance, in standard binary system we would have $$x' = 2x + s$$, what is optimal for $$\Pr(s=0) = \Pr(s=1) = 1/2$$ probability distribution. Using $$x' \approx x/p$$ rule for a general probability distribution is the base of asymmetric numeral systems family of entropy coding methods used in data compression, whose state distribution is also governed by Zipf's law.

Is it original research? Jorge Stolfi (talk) 01:22, 9 May 2023 (UTC)

Add a Non-Logarithmic graph (in the intro to the page preferably)
There is a reason the most popular youtube video on Zipfs law did not use a logarithmic graph...When you graph Zipfs law in Log scale,,,its just a linear graph...this is not understandable at a glance. A basic 1:1 graph is understandable without having to read the scale on axis (most don't even use axis its so easy). A exponential curve, sharply upwards at the end, is more understandable to the common people than anything with a logarithmic scale.

I ask we add 1 simple graph at the beginning so that a commoner can understand the curve easily.

Non-logarithmic scales are readable instantly to everyone. (logarithmic are only readable to educated people, and even they have to check the axes) Diox8tony (talk) 17:34, 5 June 2023 (UTC)

The first plot seems to be wrong.
Frequency as a function of frequency rank should be monotonically decreasing, which is not the case in the first plot of "word frequency vs frequency rank in War & Peace" (this plot --> https://en.wikipedia.org/wiki/Zipf's_law?/media/File:Zipf%27s_law_on_War_and_Peace.png#/media/File:Zipf's_law_on_War_and_Peace.png)

This plot isn't extracted from the source cited in caption (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4176592/pdf/nihms579165.pdf) but it is similar to other plots shown in this source. However all of the plots in this source are NOT monotonically decreasing, so it seems that those aren't a "Frequency as a function of frequency rank" plots.

I've never edited wikipedia before so I don't feel confident enough to propose the deletion of the plot. Qchenevier (talk) 15:13, 13 August 2023 (UTC)

Genlangs
I have removed the section on genlangs. The only citation was the original paper, which is neither notable or peer reviewed, and is of extremely low quality. Tristanjlroberts (talk) 16:42, 23 August 2023 (UTC)

title: Talk:Zipf's law