# Talk:Formal science

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## Is this term on use

Jon Awbrey 06:30, 12 January 2006 (UTC)

Is this term in use? Tom Harrison Talk 20:35, 13 August 2006 (UTC)

Yes, the formal sciences, math, logic, topology, etc.. exist.
Applied math is different from theoretical math. The math we all are most familiar with is applied math. At the university level you can choose if you want to work with math as a theoretical subject, or work with applied math. A lot of people think that the algebra you learn in school is the only existing mathematical model, and that it is infallible. The fact that it is only one of millions of theoretical mathematical models is taught at the university level. Roger491127 (talk) 14:40, 13 January 2008 (UTC)
The first line in the history section, "The study of formal science began much earlier than natural science..." is very questionable, and no reference is given. Considering that you first need to develop the basic terms and operations in applied math before you can start building more complex and theoretical math models it is highly unlikely that the development of formal sciences could begin before applied math like measuring land areas had been developed. Roger491127 (talk) 16:11, 13 January 2008 (UTC)

## The term "Formal science"

I've never heard the term "Formal Science" before reading it on wikipedia, and I am a pure mathematics graduate. Personally I rather like the term; it is a sensible classification to distinguish mathematics and related subjects from the empirical sciences (it would perhaps be more correct, though unfortunately some would find it offensive, to simply refer to logic, computer science, linguistics etc. under the general heading of "Mathematics"; ironically, I'm guessing the term "Formal Science" came about specifically through lack of a place to put mathematics in the academic spectrum, but was then defined to be basically synonymous with mathematics). However, opinions aside, this page really shouldn't exist (or, at least, the term shouldn't be mentioned on the "Academic Disciplines" page, which is where I found this page) unless it is in common usage. Can anybody find any universities with a "Formal Science" department? Whether or not the term makes sense, it shouldn't be part of wikipedia unless it has been used elsewhere (much like with "Dharmic religion"). 217.155.61.70 (talk) 13:26, 18 March 2008 (UTC)

This term shouldn't exist...!?
• C. West Churchman (1940), Elements of Logic and Formal Science, J.B. Lippincott Co., New York.
• Bernt P. Stigum (1990), Toward a Formal Science of Economics, MIT Press
And books like this shouldn't have been named like that? No, Wikipedia has the task to clearify terms, not to eliminate them. -- Mdd (talk) 16:07, 18 March 2008 (UTC)

A google search gives 34,300 hits for "Formal Science". By comparison there are 4,420,000 for "Natural Science". I'm not debating this terms existence, rather, I'm questioning its use as a classification of academic disciplines (which would make "Formal Science" to mathematics what "Natural Science" is to physics; the term does not appear to be on that same level). I will post this concern on the academic disciplines page however. For this page, I suggest it be rewritten to reflect the terms actual usage; it appears to be a nick-name for subjects with a mathematical approach, and not a historical discipline. The history section is unnecessary unless there are any historical uses of the term (link to history of mathematics instead). Similarly the overview serves no purpose if it is no different than an overview of mathematics. 217.155.61.70 (talk) 12:59, 19 March 2008 (UTC)

I think your misreading the historical section. It's not only about mathematics. It refers to mathematics, logic, statitics, computer science and information theory. -- Mdd (talk) 13:13, 19 March 2008 (UTC)
All these subjects can be considered mathematics. It is only in more modern times that they have been seperated (and still in modern times, I can't find a single university that uses the term). Fermat's work on probability was considered mathematics, as was Turing's theory of computation. There is certainly nothing from ancient India, Egypt or Mesopotamia that can be considered seperate from mathematics. If you wish to have a history section I can only reccomend it describe the evolution of these subjects into seperate disciplines, which certainly did not happen in ancient times (you're talking late 19th century at the earliest, with Frege). 217.155.61.70 (talk) 15:23, 19 March 2008 (UTC)
I am not surprised that you as pure mathematics graduate states that everything is mathematics. But your wrong that only in more modern times these fields have been separated... and all originate from mathematics. It's more the other way around. Symbolic mathematical logic and set theory originates from logic. Scientists like John Venn who are now considered to be among the greatest mathematicians of all time, but in it's own time he was considered to be a logician.
I do think that this article itself explains little about what the term "formal science" means and in which context it is being used. This article in it's current state gives hardly any knowledge of formal science. It is written as a essay. I have written the article on formal science in the Dutch Wikipedia based on three quotes. Maybe I should rewrite this article. -- Mdd (talk) 19:29, 19 March 2008 (UTC)

I don't think Statistics, to take a case I'm more familiar with, can be considered an area of Mathematics, despite Probability Theory's being its mainstay. The controversies in the foundations of statistical inference are not disputes about results in Probability Theory, but about other principles ('axioms' perhaps) that guide its deployment. E.g. the Bayesian view that rational degree of belief in different values of a parameter can be represented by a probability distribution, the principle that inference should be conditional on the observed value of a nuisance parameter - these aren't part of Mathematics. Nor are they empirically testable - could this be what distinguishes a formal science from a natural science that makes heavy use of mathematics ?

Incidentally, on a page on UK degree courses, Formal Sciences is listed as a category : http://www.ukuniversities.net/Universities/Programs/Formal_Science_Programs.html Scortchi (talk) 13:56, 6 October 2010 (UTC)

## Never heard of it

I have never heard of "formal science", until now. Google trends does not even have enough hits for it to generate a search. Natural science is a famous term, by comparison. It is misleading to even have a link from "natural science" to "formal science", which is how I discovered this page, because one is extremely well known and the other hardly anybody has ever heard of.

Besides, "formal science" is a strange term to start with. "Abstract science", or "science of the non-spatiotemporal" would be better. —Preceding unsigned comment added by 86.9.9.235 (talk) 20:45, 9 July 2008 (UTC)

"Formal Science" gives a google rate over 40.000 hits. -- Marcel Douwe Dekker (talk) 20:49, 9 July 2008 (UTC)

## Some early uses

The term is really from the history and methodology of science. According to Google Scholar, the oldest uses of the term in the sense of this article are from the mid C19th, and I found a little on the following texts in particular:

• Sir William Hamilton, 9th Baronet, 1860. Lectures on metaphysics and logic. Ed. Henry Longueville Mansel & John Veitch, pub.Gould and Lincoln — Describes logic, studying the agreement of thoughts with other thoughts, and mathematics, the science of quantity, as the two formal sciences, concerned with formal truth.
• Thomas Erskine Holland, 1880. The Elements of Jurisprudence. Clarendon Press — Defines jurisprudence as "the formal science of positive law", ie. the study of the inferential commitments of legal rules.
• Bernard Bosanquet (philosopher), 1895. The essentials of logic, being ten lectures on judgment and inference. Macmillan and Co. — defines formal science as that part of science that attends to the form of things, rather than their content. Argues that all science is formal, at least in part, because all science treats universals.
• Horace William Brindley Joseph[1], 1916. An introduction to logic. Clarendon Press — Argues that logic is not purely a formal science.
• Susan Stebbing, 1930. A modern introduction to logic. Methuen — Contains discussion of the scientific method, the status of logic and mathematics as sciences.
• Wilfred Sellars, 1947. 'Pure pragmatics and epistemology'. In Philosophy of science, ed. William Marias Malisoff, University of Chicago press — contrasts empirical to formal science, talks of empirical aspects of formal sciences (eg., marks on paper), and formal aspects of empirical sciences (eg., methodology).
• C. West Churchman, 1940. Elements of logic and formal science. Lippincott — Subject of book is logic and the philosophy of the formal sciences, contains discussion of how geometry became a formal science by gaining theoretical models.

Note that Bosanquet, in particular, uses formal science in a way that is different to the article: he is interested in the method, rather than the object, of the science. Some thoughts: first, the term seems to have been more used in the past than it is today, although I think that the Foundations of the Formal Sciences conference series has begun to change that; second, the term is useful, clarifying what is going on in scientific methodology, and in particular, observations such as those Sellars made are awkward without the terminology; third, I think the Google results mentioned above do, if anything exaggerate the frequency of this term: I saw quite a few occurences such as "formal science education" that was not about eduction in the formal sciences, while alternate usages for natural science seem rarer. — Charles Stewart (talk) 14:06, 3 June 2009 (UTC)

## Who wrote this? It is bull.

"The formal sciences are built up of symbols and theoretical rules."

... should be changed to:

"The formal sciences are built partially from symbols and theoretical rules."

"The formal sciences can sometimes be applied to reality"

...should be:

"The formal sciences can be applied to reality"

", and, within certain limitations,"

...doesn't need to be there

"People often make the mistake of confusing theoretical systems with reality"

Often?!! WTF? NOBODY CONFUSES THEORETICAL SYSTEMS WITH REALITY!!!

", applying theoretical models as if they represent reality perfectly"

WHO DOES?

", or believing that the theoretical model is in fact the reality."

HOW CAN A MODEL BE REALITY?

"The difference between formal science and natural science is that formal science starts from theoretical ideas"

No it doesn't.

"and leads to other theoretical ideas through thinking processes"

Yes it does

"One can never learn anything empirical from studying formal sciences alone."

I can. Get a mic plug it into your computer, convert the wave using a ADC and read the amplitude. Get a computer program and test for errors.

"One can never prove anything empirical through the use of formal sciences."

Again, using the same signal, this time recording a bat, do a fast Fourier transform to prove a bat squeaks above 1000 Hz.

"Applied mathematics is to try to apply some theoretical mathematical model to reality."

Not always. In fact, the majority of the time the models are made using previous data and observation.

"It is possible within certain limits and with certain restrictions and with a certain limit of precision."

Very small limits, minute restrictions and the best objective precision.

"If the map and the reality do not fit it is the map which is wrong, not the reality. A map is a theoretical representation (model) of reality."

This is final point is correct.

SOMEBODY SORT THIS ARTICLE OUT! SOUNDS LIKE SOME SOCIAL SCIENTIST HAS CRAPPED ALL OVER IT!

You did not sign your text so I do not know who you are, but I wrote the text which made you so upset. You should have left it as it was, this article is in a much worse shape now than when I wrote that text. Read the rest of my answer at the bottom of the page. Roger491127 (talk) 23:28, 22 October 2010 (UTC)

## Who wrote this? It is bull... I rewrote a lot of it

Sounds to me like it wasn't a social scientist which crapped all over it but a natural scientist. Made me laugh as whoever did it completely had no knowledge whatsoever in the field. They were just throwing cheap disses at logic and reason, which of course, they rely so heavily on. --81.129.211.63 (talk) 06:21, 13 November 2009 (UTC)

## Formal Sciences Vs Linguistics

Linguistics is a social science with some aspects being formalised, while Statistics and Computer Science are fundamentally formal sciences. For example, Computer Science could even be considered an investigation into mathematics itself (meta-mathematics), and is centrally involved with its formal aspects. IT and Software Engineering, for example, aren't computer science even if such topics are usually taught in some CS degree programmes. CS continues to be most thoroughly founded on topics such as Godel Theorem, Church-Turing thesis, Set Theory, Asymptotic Analysis, P vs NP -- i.e. topics that are essentially pure mathematics. Linguistics, on the other hand, is a social science where some topics within the discipline have become formalised. We could just as easily argue that Physics is a formal science because much of it has become formalised. Moreover, some of it is purely mathematical, e.g. string theory is only falsifiable insofar as quantum mechanics and relativity are falsifiable, and as such are not even scientific theories (i.e. do not use the scientific method). They are essentially mathematical theories. As such, I've edited the last line of WP:Lead to make this distinction clearer. However, linguistics probably ought to be removed from here altogether. Even in Wikipedia it is listed under social sciences while the listed formal sciences are Mathematics, Logic, Computer Science and Statistics. Rlinfinity (talk) 13:58, 30 May 2010 (UTC)

Methinks you aren't a linguist. The reason that the article mentions "some aspects of linguistics" is not to imply that any formalization of an object of study can lead to the study's inclusion as a formal science, but rather that the object itself--here, language--and in turn the study as a whole display characteristics required of a formal science, namely that, as Einstein said, it's governed by incontrovertible, non-empirical laws. Yes, much of linguistics strives to understand natural language in a social science fashion, but much of it abstracts away from its application (in the same way as mathematics), to form a set of ideas that exist independent of reality. This goal is in part to discover the true boundaries and properties of broadly-construed 'language'--and to do so results in a mathematical, logical abstraction that does not arise, but just is. As one prominent example, context-free grammars fell out of linguistics. Among other areas topics studied under linguistics as a formal science, formal language theory is an important one. 66.59.249.107 (talk) 12:25, 1 December 2010 (UTC)

## Proposed restructuring

The sections Theoretical models and, in particular, Differences with other forms of science appear very weak to me, almost totally unreferenced (the one reference is not from a reliable source), with spotty and unbalanced coverage, and written like a section of a sophomore essay. I propose to completely scrap the section Theoretical models and to replace the content of Differences with other forms of science by that of the current section Overview. I could be bold and just do it, but before subjecting the text to such an upheaval I'd like to hear first if someone has a better idea or other suggestions for improvement.  --Lambiam 23:17, 21 August 2010 (UTC)

I agree - those sections are not at all clear. The section 'Overview' is good, though I wonder if it could be made more neutral from a philosophical point of view : replacing 'factual' with 'empirical' & 'real' with 'actual', as here's perhaps not the place to be asserting that "2+2=4" is not a fact or does not describe the real world. Scortchi (talk) 12:36, 6 October 2010 (UTC)
So done – but I've left "real" alone, as "real world" is a standard notion, and "actual world" would appear to mean just the same.  --Lambiam 18:26, 8 October 2010 (UTC)

Good, & thanks - the article looks a lot better now. I had wanted to put 'actual' instead of 'real' because they *are* slightly different (see modal realism), but then I thought most readers wouldn't pick up on the difference in any case. For the moment I'm just changing 'fact' to 'contingent fact' - if you think all facts are contingent it's merely redundant; if you think some are necessary it makes the sentence right. Scortchi (talk) 00:23, 9 October 2010 (UTC)

## Relevance of rationalism to the formal sciences

I removed the entry Rationalism from the "See also" section; rationalism is a philosophical position within epistemology that holds that knowledge can and should be acquired by pure reason rather than empirically. Since this epistemological distinction is meaningless in the methodology of formal sciences (like in the chestnut of an engineer practising number theory: "3 is prime, 5 is prime, 7 is prime, 9 *experimental error*, 11 is prime, 13 is prime: odd numbers are prime"), this is utterly irrelevant here. However, it was almost immediately re-added (without explanation). Why?  --Lambiam 13:56, 22 August 2010 (UTC)

The history of rationalism helps us to keep in mind that the premises of deductions shold be verified by scientific experiments. I think it is also helpful to the development of formal science. CES1596 (talk) 15:04, 24 August 2010 (UTC)
To take a random example of a deduction from a formal science, the proof of Lindelöf's lemma proceeds from the premise that some subset S of the real line is an open set to deduce the conclusion that S is a countable union of open intervals. Please enlighten us what kind of experiment might be conducted to verify this premise.  --Lambiam 23:44, 24 August 2010 (UTC)
You can make and apply any arbitrary rules if you do not have to verify them. It is what has been done in a number of dictatorships. The theory of real numbers is accepted because it is consistent with the real world. CES1596 (talk) 14:54, 27 August 2010 (UTC)
I'm afraid your understanding of the acceptance of mathematical theories is based on a grotesque misconception. See the quote by Einstein in the article. Here is another quote form Einstein: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. You also evaded my question: What conceivable experiment in the real word could be used to test the premise that some subset of the real line is an open set?  --Lambiam 18:30, 27 August 2010 (UTC)
You can test it by using a compass and a ruler. The consistency of the theory of real numbers is equivalent with the consistency of the Euclidean geometry. CES1596 (talk) 11:47, 29 August 2010 (UTC)
Apparently you have no clue what "open set" means, otherwise you would (I dare to hope) not offer such an utterly nonsensical suggestion. In general, it is advisable to make sure one understands a question before answering it, if only for not appearing a fool. Additionally, it is possible to do analytic geometry entirely with the constructible numbers, whose topology differs in essential ways from that of the real numbers, so it follows that you cannot test statements about the topology of real numbers with compass and ruler.  --Lambiam 12:59, 29 August 2010 (UTC)
I mentioned ruler and compass constructions for bisecting intervals recursively on the real line. CES1596 (talk) 16:17, 29 August 2010 (UTC)
That is not relevant for the premise of Lindelöf's lemma, but let us take a simpler example: the property of the real numbers that between any two given different numbers there is a third real number. Is your claim that we can test that by bisecting an interval of length, say, 10−35 meters, using an actual physical ruler and compass? That must be based on a novel insight into the nature of reality.  --Lambiam 10:28, 30 August 2010 (UTC)
I just gave an answer to your question. Anyway, I think Einstein referred to the laws of nature as the laws of mathematics and observation as reality. The certainty of logic depends on the centainty of the inductive reasoning based on observation. CES1596 (talk) 17:32, 31 August 2010 (UTC)
Apparently you are not able to see the fundamental difference distinguishing the formal sciences from the natural sciences. I give up.  --Lambiam 09:13, 1 September 2010 (UTC)

## Name changed to "Formal sciences"

I agree with the remark of Roger491127 above that the name "Formal science" was wrong. Just like "Humanities", "Formal sciences" is used in the plural. Although you can say that "mathematics is a formal science", in that sentence "formal" is just an adjective applied to "science", like "exacting" is in the sentence "mathematics is an exacting science". Therefore I've changed the name to plural.  --Lambiam 07:25, 30 November 2010 (UTC)

Why would you compare it to humanities? Its more logical equivalent is 'natural science,' which is the name of its article; 'natural sciences' redirects to it. The opposite happens to 'social science' and 'behavioral science.' I see both as correct, depending on what you are referring to: the general idea behind those sciences embodied by 'formal science,' or the group of sciences that constitute 'formal sciences.' The former is not wrong. 66.59.249.107 (talk) 12:37, 1 December 2010 (UTC)
When enough people do not consider it wrong and start using the term in this sense in the singular, then it becomes right. As far as I can see, they are still a small minority. "Natural science" feels right to me, because it is concerned with the study of nature, and one of the meanings of "natural" is "pertaining to nature", as in, e.g., "natural experiment". "Formal" can mean "pertaining to form", but the object of study of the formal sciences is not form.  --Lambiam 21:42, 1 December 2010 (UTC)
Given that few people seem to have even heard of it (either "formal science" or "formal sciences"), it's unreliable to base this on "what people say," so nobody can say that any group is "in the minority" (unless you have a study up your sleeve). Furthermore, I was making the point that "formal sciences" is not wrong, but represents a different use of the term appropriate for specific contexts, in the same vein that "the team are" and "the team is" are both correct in British English (though number marking on the noun here has no morphological reality--only on the verb--you see how the semantics dictate different forms for different contexts, depending on speaker intent). And just as "formal sciences" feels right to you, "formal science" feels right to me--and until you can prove that the former is more common, your reasoning is unjustified. "Natural science" and "natural sciences" are both right, but occur in different contexts. Also, your point about "formal" also meaning "pertaining to form" is irrelevant; "natural" has some 40 meanings in the dictionary, though only one of them is interpreted in the case of its combination with "science." The point I'm making here is that by choosing to name this general area of study "formal science[s]," we select for a specific meaning of "formal" to lend a compositional meaning for the larger term "formal science," and so decomposing it to argue against a specific form makes no sense, especially when you yourself just argued that linguistic description--not prescription, based on personal logic--is the basis for the terms.
And again, I'm not arguing for a change in the title (who really gives a ****?), but as a linguist by education and trade, I felt compelled to refute the assertion that one of the forms is wrong. It isn't. 66.59.249.107 (talk) 08:07, 5 December 2010 (UTC)

## Unsourced tag

I added an unsourced tag because the there is only one cite in the article and that is for a quotation. I'm having trouble verifying the definition given in the lead. The most clear and concise definition I found in a Google books search was that the formal sciences consist of mathematics and logic, though this definition is probably obsolete. The other sources I found either used the term without definition or used different versions according to the author's field. If this is a generally recognized modern definition then please add a reference to that effect, otherwise the article should be either renamed or deleted.--RDBury (talk) 00:47, 4 July 2011 (UTC)

Yes, it needs more references. I had thought the term originated with the Vienna Circle, but some of the references here in the talk pre-date them (& perhaps whoever found them might like to integrate a few into the article itself). Still, I don't think modern usage has changed since the VC, though the list of examples is now a bit longer. I'm going to have a look this week. Scortchi (talk) 23:13, 27 September 2011 (UTC)

## Differences from other forms of science

There've been a number of changes in 'Differences from other forms of science' blurring the distinction between formal & empirical sciences: e.g. 'formal sciences do not usually involve empirical procedures'. Overall they've made things less clear - do just a few formal sciences use empirical procedures or do all of them occasionally ? - & introduced contradictions - formal sciences only 'sometimes lack empirical content' but 'contain no synthetic statements'.

What's the motivation for these changes ? If it's doubt that one of the examples of formal science given fits the definition, it would be better to discuss this specifically. If it's philosophical qualms about the analytic/synthetic distinction, it would be better to note lower down that if the distinction between analytic & synthetic statements is one of degree rather than of kind, so is the distinction between a formal & a natural science, & to put in a link to Quine. I'm planning to put this section back how it was. Scortchi (talk) 14:47, 27 September 2011 (UTC)