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I have always been wondering

why the term "model" is used by mathematicians (especially in mathematical logic) in a conceptually different (even opposite) way than it is used by other scientists, e.g. physicists, biologists, chemists, economists etc. And when has this terminology first arisen?

In essentially all of natural sciences we have a "world" (our physical world, a living being, a market...) which hosts specific instances of certain "phenomena" (that involve "objects" of some kind, and "relations" between them), and the goal of the theory is to provide a model that describes that phenomena. In this case, a model is understood to be some kind of conceptual construction that abstracts the common relevent properties of several instances of the phenomenon in question and puts them in a rational (mathematical or not) framework that enables us to draw consequences and predictions about the phenomenon itself. Think e.g. of the "Standard Model" of particle physics.

Also mathematics has a "world", populated by objects (e.g. groups or ordered fields), between which some relations hold. The goal of a mathematical "theory" (at least from the perspective of formal theories in the sense of mathematical logic) is to provide a simple "model" (mind the use of quotation marks) that describes all the instances of certain "phenomena", and it accomplishes this task by a list of axioms (e.g. the axioms of group theory) that incapsulate the relevant properties of the objects in question (e.g. being a group).

It is possible that new "species" of the same kind of objects are found, that is, they verify the axioms, i.e. they fall under the description by the same "model". A natural scientist would call such an object an incarnation (manifestation, realization, example, explicitation, instance...) of the "model" provided by the axioms. Mathematicians, on the contrary, call it a model (here in the tecnical sense, hence without quotation marks) for the axioms.

Isn't it strange?

(Edit: there is also another way the word "model" is used in maths, as in "Weierstrass model" or "Néron model", in which cases it is essentially sinonimous with "normal form". This latter use of the word seems to me more consistent with the general natural sciences use)

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I don't find it strange. Even in ordinary language, the same word can mean different things. By the way, I think people in applied math use the word "model" in the same way as the rest of science. –  Donu Arapura Oct 15 '10 at 15:28
A physicist doesn't use the word model in a way that's different from a logician. A theory in logic abstracts certain essential properties of some objects arising in mathematical nature. After the abstraction, one discovers other models of the theory. Similarly, a physicist abstracts essential properties of one big object (the universe) or some subobjects and builds a theory (at this point called something like 'physical principles') of which the object in nature is a model. The goal is then to build a model of this theory in mathematical nature. –  Minhyong Kim Oct 15 '10 at 15:44
@MinhyongKim: I don't agree with your point of view, as you seem to suggest that the use of the term "model" in natural sciences is somehow borrowed from math logic, at the point that I think a physicist would not even understand your argument... but we should ask a physicist :) –  Qfwfq Oct 15 '10 at 16:18
I agree with MinhyongKim. In any observation of the natural world we model that world using our senses. We filter out detail and 'explain to ourselves' in terms of our world view (i.e. our theory of the universe). In other words we cut down the external observational world to a model that fits our theory of what it is. This may be linguistic, use analogies or mathematical equations. That seems to me to be very near to the sense that a logic model is an instance of the axioms; it may have other properties but those are not being looked at by the theory. Semantics and Syntax are linked! –  Tim Porter Oct 15 '10 at 20:00
@Minhyong, @Tim: the way I've heard physicists use “model” is different from what you describe. They wouldn't say “this object in nature is a model of the theory”; instead, they’d say “this theory is a mathematical model of the object in nature”. As I understand it, this was what the OP meant about their usage being “even opposite” to the mathematical one. –  Peter LeFanu Lumsdaine Oct 15 '10 at 23:31

5 Answers 5

No, it's not strange. Understand why it's not strange, and you understand the essence of one of Frege's great innovations in logic: the so-called "linguistic turn", in which he taught us to shift from trying to study reasoning in and of iteself, to studying the language we use to describe arguments.

So, the "natural world" of mathematical logic is syntax. The rules governing logical syntax give rise to very complicated combinatorial objects, and quite difficult to study. (Structural proof theory, the study of proofs, is correspondingly a much less well-developed field than model theory.)

In order to understand the rules governing logic syntax, one approach is to to start using models. That is, we let go of working directly with syntax. Instead, we try to find mathematical structures which mimic salient parts of the syntax, but which we (hopefully) understand better than the raw grammatical rules governing the use of syntax. A model is an idealization of a logic, which quite purposefully hides the rules of inference from view -- this is why we say model theory is the study of provability, rather than of proofs.

Models are not the actual theorems and proofs we write in our notebooks. They are, quite literally, models of those proofs and propositions, but which are simpler and better-behaved.

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Personally, I still find the logician's use of the term quite backwards (as for most non-logicians the "world" is made up of "actual mathematical objects" -with a background theory understood such as ZFC- and not, or not only, every single formal theory such as PA or ACA0). Nevertheless, I found your answer very illuminating on the logician's perspective. –  Qfwfq Oct 15 '10 at 16:03
@unknown: "Most non-logicians" means just the non-logicians among mathematicians. Thinking that "actual mathematical objects" exist is the attitude which differs from the views in rest of science; this is genuine to (some) mathematicians, the rest of the world would attribute a bigger degree of reality to observations and our verbal descriptions thereof, i.e. what becomes the axioms in logic. –  Peter Arndt Oct 15 '10 at 20:54
This reminds me of a lovely observation from Thorsten Altenkirch. When you teach logic to maths students, they think of syntax as alien and abstract, and models as nice concrete things — look, they're ordinary mathematical objects, just sitting there! When you teach logic to computer science students, the mathematical models are the scary abstractions, and the syntax is this nice concrete stuff you can really get your hands on — look, I'm playing around with the formulas right there in Agda (or Coq, etc.)! –  Peter LeFanu Lumsdaine Oct 15 '10 at 23:35
I agree that the two uses of the word are very dissimilar; but I don't see that there's anything wrong with this. Natural language is rife with 'ambiguity' that rarely causes confusion in practice. Why should technical language be any different? –  Saul Glasman Oct 16 '10 at 4:17
The natural world of <i>mathematical logic</i> might be syntax, but the natural world of <i>mathematics</i> is surely not. We care about the natural numbers for their own sake; we care about Peano Arithmetic only because it captures some of the important properties of the natural numbers. Likewise, physicists care about physical reality in its own right, and care about their models only because they capture some important properties of that reality. In that sense, it's PA, not N, that's analogous to a physical model. –  Steven Landsburg Mar 30 '11 at 15:31

It may seem a bit backwards, but one could try to look at it the other way around: pretend the axioms and theorems are the things that we observe. We don't really observe the field $\mathbb R$ of all real numbers directly, but we sort of observe phenomena like $$ (\forall x)(\forall y)(x+y=y+x),\quad\text{and }(\forall x\ne 0)(\exists y)(x\cdot y=1). $$ Then we make models such as $\mathbb R$, and maybe $\mathbb Q$ or other algebraically closed fields of characteristic 0, to explain what we have observed. If we initially use the model $\mathbb Q$ but later "observe" that there seems to be a completeness property of numbers, then we may adjust our model and use $\mathbb R$ instead.

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I like your interpretation. –  Qfwfq Oct 15 '10 at 15:55
I am not sure that I have ever observed the field of real numbers, even indirectly. I have used the field of real numbers as a model that may or may not fit the situation I am studying. –  Tim Porter Oct 16 '10 at 19:50

So, I'm a "natural scientist" and I feel slightly misrepresented.

I think the idea of a model as an abstraction is probably not quite right. I consider it to be more like metaphor. For example, I am personally a big fan of kinematic models such as the ones created by the cyberneticists. I case I can think of is where a model of economics may consist of a series of buckets of water with various sized holes and gizmos to control the flow. This type of models are still alive in areas of robotics and artificial life.

I think that the mathematical idea of morphisms capture the idea that one object is a model of another (the morphism is not the model itself). It seems that you consider the model to be what is preserved between two objects, whereas I would consider one to be a model of the other if there is a preservation of something between them. Of course, whether or not something is preserved between two objects requires a third that is able to make such a judgement (this is probably the major source of distrust of theory by experimentalists).

There is no reason within the natural sciences for the model to belong to a formal mathematical system (although this is often the case). Neither is it true (except in the particular formalism of model theory) for a model of a formal system to be itself a formal system (a point famously elaborated on by Tarski). Some very nice examples of this range from the mechanical integrators and spirographs to, arguably, home computers.

Also, I quite like Bjorns example. I think it is completely consistent with what I consider a model to be.

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I don't understand your impression that logicians have distorted the word "model" by the way they use it. In both contexts (math and science), a "model" is an example of a phenomenon described by a theory. I don't think it's really required that the example be a concept.

In physics, the Ising model is an abstract system that exhibits the kind of behavior you see in ferromagnetism. Iron is also such a system, it just isn't a abstract system. Both of these are models of the theory of second order phase transitions - the description of how certain many-body systems behave under heating and cooling.

The wall clock time in hours is a model of the theory of addition, mod 12, but that doesn't make time itself an abstraction.

Fruit flies are often referred to as "model organisms" for the theory of genetic inheritance. The theory is the description, and life forms that exhibit it are models.

Granite would be a "model" of the "theory" of igneous rocks, where part of the theory says something like "it's rock that comes from cooling magma".

A person wearing a dress to exhibit how it might look is even called a "model" - they literally perform the act of wearing the dress, and thereby serve as an example that makes understanding the behavior of the garment easier than a linguistic description would. Models by definition exhibit the specific phenomenon they're modeling.

Yeah, some models are abstract, often so that they're simple enough to be understood (and sometimes you call those "toy models".) But being an abstract, conceptual object is not essential to being a model of a theory.

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@ <<In physics, the Ising model is an abstract system that exhibits the kind of behavior you see in ferromagnetism. Iron is also such a system, it just isn't a abstract system. Both of these are models of the theory of second order phase transitions>>. I understand your point; I may be mistaken, but I think this doesn't represent the general vocabulary of physicists. A physicist (I presume) would say that (...) –  Qfwfq Oct 16 '10 at 1:03
(...) "the theory of second order phase transitions is a model (partly not a mathematical one, but still a model) of the natural phenomena you can observe happening in metals"; and "the Ising model (which is a fully mathematical one) is a more restrictive case of the above model". –  Qfwfq Oct 16 '10 at 1:03
  • yes, it is strange, because one expects words to have similar meaning even in different contexts (but as others have noted, whether this is strange or not, there are all sorts of words that have multiple meanings, and even contradictory ones).

  • yes, I agree in your assessment of how model has two, not contradictory, just oppositely directed meanings. In common language, natural sciences, and statistics, a model is an abstraction, where incidental details are removed, leaving just the bare bones abstract thing of what you care about (a model airplane, a model of population dispersal, a regression model predicting a value from a small set of variables). In mathematical logic, it is the other way around; a model is an -example- of the syntactically presented axioms (the permutations of 3 items is a model of the group axioms, a valuation (an assignment of boolean values) is a model for a propositional statement.

In short, in mathematical logic, there is the abstract theory that has a model as an example, the model fits the theory. In the rest, the model -is- the theory and the phenomena of the world are the examples (which the theory/model are trying to fit).

When this divergence of use started and by who, I can't say. But the logical use is certainly later. Who first started using the concept (but not necessarily the usage) in logic? Was it Goedel? How about Hilbert in 'Foundations of Geometry' where he shows independence of individual axioms by giving different models satisfying the rest (i don't know what terms he uses in the original or if they were common usage at that time for the concept).

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