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This question is partially related to How does "is" work? and What is the difference between the "is" of predication and the "is" of identity?, but more specifically it concerns statements about physical quantities and measurements.

[Question heavily re-edited because unclear]

Consider the statement "the instantaneous velocity of body B is 1 m/s". This is usually expressed in a mathematical form like "v = 1 m/s" or similar.

The "=" in that form seems to suggest that this statement expresses a relation. But is this really the case? or does that statement express a predicate?

The question is also related to the fact that we can consider the statements

"the instantaneous velocity of body B is 1 m/s"

and

"the mean velocity of body A is 1 m/s"

and deduce

"the instantaneous velocity of body B is the same as the mean velocity of body A

All three statements – which are perfectly fine – can be expressed in deceivingly similar mathematical forms: "v = 1 m/s", "u = 1 m/s", and "v = u".

The last expression is particularly nasty, because "=" or "is the same as" really means "has the same value as", and not "is the same (thing) as". The latter would obviously be false, because mean and instantaneous velocity are different notions, and moreover they refer to different bodies.

By the same token, "the instantaneous velocity of body B is 1 m/s" seems to really mean "the instantaneous velocity of body B has the value 1 m/s".

Clearly(?) the statement "v = u" expresses a relation ("has the same value as"), and the relation is symmetric, reflexive, and transitive; an equivalence relation.

But does "v = 1 m/s" also express a relation? It seems to me that it doesn't. The chain of reasoning {"v = 1 m/s", "u = 1 m/s", "v = u"} seems to use transitivity and symmetry, and suggests that "=" is the same along the chain, and it's therefore a relation.

And yet I think that "=" denotes different things in the chain. It's mixing apples and oranges. And yet, graphically at least, it seems to work.

My questions are:

  • Does the statement "v = 1 m/s" (or "the instantaneous velocity of body B has the value 1 m/s") express a predicate? Or a relation? Or something else? Or is the intepretation arbitrary? (I would be inclined to see it as a predicate, but in a second-order logic with an infinity of such predicates, over which we can use quantifiers.)

  • How does an inference chain such as {"v = 1 m/s", "u = 1 m/s", "v = u"} work, if the "=" has different meanings in it? (eg if it expresses a predicate in the first two statements and a relation in the last.)

  • Can you suggest any literature on this issue, in relation to measurement statements in science?


I can add that a similar question arises in probability theory, which typically has statements about "random variables" of the form "X = x". This would mean "the quantity X has value x". But such an interpretation has lead to a great debate about causality and correlation. The problem is that more precisely "X = x" can mean two things: "X has been observed to have the value x" and "X has been set to the value x". These two very different statements have hugely different consequences, for example if they appear in the conditional of the probability of another statement.

[Note: I'm a physicist and this question is very important for my field, so I'd like to keep some science- or physics-related tag for this question.]

pglpm
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    When we say that "the velocity of body B is 300 km/s" we are asserting that the value of the magnitude velocity is ... – Mauro ALLEGRANZA Jun 02 '20 at 07:53
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    When we write the mathematical formula "v = 300 km/s" we are asserting the same fact: the value of the "function" *velocity* (space/time) for a given body. – Mauro ALLEGRANZA Jun 02 '20 at 07:54
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    In both cases, "is" is **not** apredicate. – Mauro ALLEGRANZA Jun 02 '20 at 07:55
  • I'm not sure I agree. Suppose you check the colour of a ball that was hidden in a box. You say "this ball is red". Isn't that a predicate? But it's also the result of your measurement. According to your view "red" is the value of the function "colour", and it isn't a predicate. – pglpm Jun 02 '20 at 07:57
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    Why is my original "physics" tag inappropriate? – pglpm Jun 02 '20 at 08:01
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    In physics, there is no doubt as to the fact that it is an " is" of identity. But there is actually a metaphysical issue which is adressed is Aristotle's categories. You may have a look at the SEP entry on " Aristotle's Categories" and on " Aristotle's Metaphysics" ; also, Marc Cohen's personal website. For Aristotle , " Peter is 6 feet tall" attributes the predicate " 6 feet tall" to Peter. –  Jun 02 '20 at 11:05
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    @RayLittleRock but consider the statements "v = 1 m/s" and "u = 1 m/s", where v refers to the *instantaneous* velocity of some body and u to the *mean* velocity of some other body. We can deduce "v = u", but the meaning of this "=" is different. The first means "has the value", the latter means "as the *same* value as" – which is a different kind of statement. Moreover, "to have the same value" is very different from "to be identical with". In fact, instantaneous and mean velocities are different notions, with different definitions. They are not identical. – pglpm Jun 02 '20 at 11:22
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    @RayLittleRock thanks a lot for the references! I'll check them. – pglpm Jun 02 '20 at 11:24
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    Suppose I say, using the language of function ( for as M. Allegranza points out, we are dealing with functions here) " Color ( my car) = white " and " Color ( you car) = white". In can conclude that " The color of my car is the same as the color of your car " , " is the same " meaning " is identical to". So the color is the same. But, the *cars* are *equivalent* : they belong to the same equivalence class under the relation " x has the same color as y ". –  Jun 02 '20 at 11:31
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    What is right in your account, it seems to me, is that the particular instance of white that is *in* my car ( the *trope* as some philosophers say) is not identical ( numerically the same object) as the particular instance of white that is in your car. Details about this in Aristotle's categories. –  Jun 02 '20 at 11:34
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    I see what you and Allegranza mean now, thank you. I notice that there's a perhaps obvious, arbitrary component in the modelling of such statements. This is why I'm also asking for literature on this topic. – pglpm Jun 02 '20 at 11:36
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    You may find things regarding the semantcs of physics in patrick Suppes, "Introduction to Logic pdf" ( in the chapter where it is deals with the axiomatization of physical theories). Also, maybe some phisically minded physicst may answer that kind of question on Physics SE. –  Jun 02 '20 at 11:42
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    I checked Suppes's text indeed – it also has a nice demonstration that "F = ma" cannot be a definition. which some physicists mistakenly believe it is. I don't remember him addressing the problem of physical quantities in such detail, but I'll recheck. – pglpm Jun 02 '20 at 11:48
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    "The ball is red" can be analyzed with subject-predicate: "Red(ball)" where "Red( )" is a predicate (a property predicate of something) and "ball" is the subject (an object of which the "redness" is predicated). – Mauro ALLEGRANZA Jun 02 '20 at 11:57
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    Maybe useful Patrick Suppes, [Studies in the Methodology and Foundations of Science: Selected Papers from 1951 to 1969 (Springer, 1969)](https://books.google.it/books?id=C4j1CAAAQBAJ&printsec=frontcover) as well as Joseph Sneed, [The Logical Structure of Mathematical Physics (Reidel, 1971)](https://books.google.it/books?id=ypruCAAAQBAJ&printsec=frontcover) – Mauro ALLEGRANZA Jun 02 '20 at 12:43
  • Thank you! both look great, especially Sneed. Please feel free to post you comments and literature as an answer. – pglpm Jun 02 '20 at 12:50

3 Answers3

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The statement "The ball is red" can be rewritten with subject-predicate form: "Red(ball)" where "Red( )" is a predicate (a property predicated of something) and "ball" is the subject (an object of which the "redness" is predicated).

In this form, there is no "is". This is the background for the assertion that, in statements like that above, "is" is not a predicate.

But "is" means also equality: a logical binary relation.

We use it ubiquitously in mathematics and physiscs: for example in equations, like e.g. 1+1=2 and F=ma (see e.g. the post What exactly is an equation?).

When we say that "the velocity of body B is 300 km/s" we are asserting that the magnitude velocity has a certain value.

Thus, when we write the mathematical formula "v = 300 km/s" we are asserting the same fact: the value of the "function" velocity (space/time) for a given body.

This means that, IMO; "v = 300 km/s" is not an equation (an identity between e.g. numbers or functions).

The formula "v = 300 km/s" is like an assignment in logical programming: "let the value of v be..." that we use in describing a physical problem, or the result of an operation of measurement.

Some useful references:

and:

Mauro ALLEGRANZA
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  • I was wondering if, when you are saying "a property predicated of something", you really want to make an ontological commitment or whether this is just an accident. I mean, one can -- as I did in my answer -- perfectly well say "Red()" is a predicate without committing to properties. Thx. for a clarification. –  Jun 02 '20 at 14:46
  • Maybe your explanation can be put as follows: we have a function *F* meaning "has the value", so we have "F(v)=1m/s" and "F(u)=1m/s", and we can also say "F(v)=F(u)". What confuses me is that "F(v)=1m/s" seems to be a meta-statement, rather that a statement within the formal system, whereas "F(v)=F(u)" is within the system. Am I wrong? And would the deduction "F(v)=1m/s", "F(u)=1m/s": "F(v)=F(u)" be within the system, or on a meta-level? And also, what are the objects in the range of *F*? – pglpm Jun 03 '20 at 10:23
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    @pglpm - as per my answer, the issue is a little bit thorny... We manage physical equations in the same way we manage "usual" equation: if we have the expression **(x-1)** and we let **x=1**, we substitute the value of **x** into the equation and we compute the result: **(x-1)[x ← 1]=0**. The same for a physical case: **Ek= 1/2 m v^2**, for a body with mass **2** and **v=2** we substitute to compute the kinetic energy: **Ek=v^2=4**. The computations are performed in the same way using the [properties of equality](https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms): – Mauro ALLEGRANZA Jun 03 '20 at 10:52
  • **x = y → (P(x) → P(y))** where **P( )** is a formula. My issue in treating the formula **v=2** as a "standard" equation is that from **m=2** and **v=2** we cannot meaningfully conclude that **m=v**, i.e. that the magnitude *mass* and the magnitude *velocity* are identical. We have to be consistent and asserting that "the value of the magnitude *mass* and the value of the magnitude *velocity* are (in that specific context) identical". – Mauro ALLEGRANZA Jun 03 '20 at 10:55
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The question seems to have both a logical aspect and an ontological one.

But first , let me try to explain how I see the reasoning that is behind the question.

(1) You are not willing to accept that instantaneous velocity and mean velocity are the same thing ( with sameness = identity). (In particular , you are not willing to admit that the instantaneous velocity of an object A be the same thing as the mean velocity of an object B).

(2) But you encounter this perfectly possible case : v = 1 m/s and u = 1 m/s with v denoting the instantaneous velocity of object A and u denoting the mean velocity of object B

(3) And here, a problem arises, because identity is symmetric transtitive. So , if you admit (2) , you are forced to say that : v = 1 m/s and 1 m/s = u , in such a way that : v = u

(4) Since the final step of (3) is apparently incompatible with your principle (1) , you want to escape this situation in the following way :

in (2) the assertions should not be expressed as equalities, but as subject - predicates sentences. The subject - predicate relation is not symmetric and transitive, so in that case, the undesired result at (3) is avoided.

What I would like to show is that there si no contradiction betweeen (1) and (2)-(3) .

ONTOLOGICAL ASPECT

  • You can first say that you do not identity a physical reality ( a force, a quantity) with its measure, its value. ( after all, a measure is only a number, but physics is not ultimately about numbers, abstract entities ; it is about concrete znd causally efficient reality)

  • But your symbols " v " and " u" denote the values of the physical realities " instantaneous velocity" and " mean velocity" . ( Meaning : the two symbols do not denote the physical realities themselves).

  • So, saying that v = 1 m/s ( identity) and that u = 1 m/s ( identity) , implying that v = u ( identity) is perfectly OK and harmless.

  • The two velocities ( physical realities) are simply equivalent in values ( as you pointed out) , while their values are really and unproblematically identical ( one and the same number).

LOGICO-SEMANTICAL ASPECT.

Note : here , one can make use of a distinction that we owe to Frege between the denotation of a symbol and its sense ( the concept that it expresses).

  • For example : " the morning star" and " the evening star" denote the same object, but do not express the same concept.

  • The statement "v = u" becomes harmless when we consider it in a purely extensional way. In its extensional reading ( the one that mathematical science favours) it simply means

the number denoted by v is the same number as the number denoted by u

  • But that does not make your principle (1) false ; because your principle is still valuable at the intensional level :

in spite of the identity v = u , there is no identity between the concept of v ( that is, the concept of instantaneous velocity ) and the concept of u ( the concept of mean velocity)

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    Thank you for this point of view. I'm not sure if we can really conflate "having/assuming a numerical value" with "denoting a numerical value". Say, "age(Bob) = 20 years". Does the age of Bob denote 20 years? or refer to 20 years? I'm not even sure I would understand what that means... – pglpm Jun 02 '20 at 13:56
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    It seems my question deals with something that hasn't been settled yet. – pglpm Jun 02 '20 at 13:57
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    you're right, the question is not an easy one –  Jun 02 '20 at 14:01
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    I appreciate all points of view and literature proposed so far nevertheless – pglpm Jun 02 '20 at 14:01
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    wow greatly extended answer! – pglpm Jun 02 '20 at 17:45
  • It seems there is a way to solve the problem by considering the instantaneous velocity as conditional mean velocity : " A has a 1 m / s instantaneous velocity " can be interpreted as " if object A continued its movement (without being accelerated) for 1 second , it would cover a 1 meter distance" . In that case, it looks less odd to say that the instantaneous velocity of A is the same as the mean velocity of B. –  Jun 03 '20 at 08:41
  • https://www.feynmanlectures.caltech.edu/I_08.html –  Jun 03 '20 at 08:43
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    It seems that the problem itself is unclear – to me too – and that we're drifting away from it. I need to edit and rephrase my question. It's clear that "v=1m/s" and "u=v" have different meanings, and that the latter doesn't mean "is the same as", but "has the same value as". It's also clear that the second is a (transitive and reflexive) relation. The question is what "v=1m/s" is. Not a relation. I still think it's a predicate, but we need a second-order logic with an infinity of such predicates and the possibility of quantifying over them? – pglpm Jun 03 '20 at 09:13
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    I've heavily re-edited the question. Maybe it makes more sense now? – pglpm Jun 03 '20 at 09:44
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    What you say about construing the sentence as a subject oredicate sentence point in the direction of Aristotelian philosophy of mathematics ( and aristotelian conception of measurement). There is a web site of an australasian Aristotelian school of mathematicians . https://www.amazon.fr/Aristotelian-Realist-Philosophy-Mathematics-Structure/dp/1137400722#reader_1137400722 –  Jun 03 '20 at 10:10
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A few thoughts:

-do all of these conceptual questions still cause difficulties if everything is expressed purely in mathematical terms and in full without abbreviations? I don't have the right typeset on my computer but, for example, I think that if you typed out the formula for mean velocity in full it would be quite clear what is meant by saying that it some case a given mean velocity is equal to a given instantaneous velocity.

-talking about whether motion is a property or a relation seems to me to be closely related to the general discussion about substantivalism vs relationism in respect of space and time; if you haven't already read it John Earman's World Enough and Spacetime is a great book on this

-re the notion of equality vs equivalence in mathmatics; I believe that there is a whole area of research on this at present, with Jacob Lurie at IAS playing a leading role. See this popular article at Quanta https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/ - This sounds fascinating, but Lurie's 944 page book 'Higher Topos Theory', however, sounds a very long way further into the domain of hardcore pure mathematics than I am ever likely to venture!

Rollo Burgess
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