An Viable Alternative To Platonism In Math?

[peoplater]

You don't think they were great philosophers, but that addresses the CONVERSE of your assertion, not your assertion. You claimed that philosophers of mathematics were not mathematicians. When it was pointed out to you that that is incorrect, you've come around to say that some of the mentioned mathematicians aren't much as philosophers. Anyway, the overwhelmingly vast number of philosophers of mathematics have been mathematicians. Wittgenstein is one example of a philospher who is usually not considered a mathematican. One of the very few. But what is the basis of your claim that he was not conversant in mathematics? In any case, the point stands that you are flat out incorrect in your original assertion, especially regarding the late nineteenth century and twentieth century. Even in earlier centuries, at least most of the famous philosophers who opined on mathematics weren't exactly ignoramuses about the subject.

You don't understand what my point is and keep pointing out that none of my assertions have any basis. In fact the basis is my personal experience and the books I have read on the matter. So here is my point. To talk about any subject, one should at least understand it and have some kind of intimate knowledge of the subject. Such knowledge comes after years and years of being immersed in the subject. Most philosophers do not have such a background in mathematics, and that is why when they "opine" on the matter, their "deep" philosophical inquiries seem quite childish. Plus, all the great mathematicians that were also philosophers, had very little to say on the subject of philosophical foundations of mathematics. If you read any of their works you will notice this. Most of their writings deal with logical foundations and there is very little actual philosophical inquiry. Most of their writing deals with trying to provide a foundation that is free of paradoxes. The little that does deal with philosophical foundations of mathematics is simple and nothing special.

[LauricAcid]

Of course I agree that people who don't know anything about mathematics are up to their necks when they pontificate about the subject. But, the vast amount of published work (I don't mean popularizations like the David Foster Wallace book and others) on the philosophy mathematics - journal articles, textbooks, et. al - is written by mathematicians or philsophers who have studied mathematics. Dedekind, Pierce, Frege, Hilbert, Russell, Brouwer, Heyting, Quine, Godel, Church, Carnap, Hintikka, Feferman, Putnam, Kripke, Dummet, Benacerraf and my favorite, Curry. One could list hundreds. And going back through the centruries. Hey, Plato was not exactly a mathematical dummy. There are entire shelves and sections of libraries and thousands of Internet pages full of philosophy of mathematics written by mathematicians. If you've not come across any of this stuff, then you need to get out a little more often.

You have an important point: too many people who don't know about math are only too willing to spout nonsense about it. But you've overstated when you say that the philosophy of mathematics is bereft of mathematicians and the mathematically savvy. And especially of philosophers in the last fifty years or so. What example do you give of a philosopher whose writings on the philosophy of mathematics are insufficiently informed? (I mean philosophers, not merely popular writers and not post-modernists, or whatever they're called.) Surely you're aware that many (most? nearly all?) university philosophy departments have so much math swirling through them that they might as well be math departments.

the books I have read on the matter.

Which books? What are the books on the philosophy of mathematics, written by philosphers, that you've read that are not sufficiently informed?

Plus, all the great mathematicians that were also philosophers, had very little to say on the subject of philosophical foundations of mathematics. If you read any of their works you will notice this. Most of their writings deal with logical foundations and there is very little actual philosophical inquiry.

That's a different complaint. You're taking exception to the focus of these mathematicians' philosophical writings. But that does not entail that they are not about philosophical foundations. Anyway, right off the bat, Quine and Putnam are two salient counterexamples. They're very much concerned with the broad philosophical picture.

Most of their writing deals with trying to provide a foundation that is free of paradoxes. The little that does deal with philosophical foundations of mathematics is simple and nothing special.

The foundational issues are the Grand Central Station of the philosophy of mathematics, which makes perfect sense. But there is fervent debate on all kinds of issues. Yes, most of them touch back to the foundational questions, but in the same way that philosophy itself keeps touching back to certain central issues. Anyway, there is so much written on the philosophy of mathematics, and by mathematically astute people, that it just overwhelms me thinking about how much I'll never even get a chance to look at, let alone read. Edited November 14, 2005 by LauricAcid

[peoplater]

That's a different complaint. You're taking exception to the focus of these mathematicians' philosophical writings. But that does not entail that they are not about philosophical foundations.

But that is my point. Their writings, the ones I know of anyway, are so mathematically flavored and use so much mathematical intuition that they can hardly be considered philosophy. Any math student after spending some time with certain mathematical concepts would come to the same conclusions.

[LauricAcid]

Okay, I hear you. A lot of the stuff is technical. I mean, you know you're in for a ride when a "philosophy" book has a three page index of mathematical symbols. But there's plenty that has been written with hardly a formula in sight. How about Putnam's terse little book? A classic platonist attack on nominalism. A great book. Or Quine? Six Dogmas Of Empiricism is a touchstone. How about Curry's moderate formalism? I'd love to hear your thoughts on his brief book. How about the logical positivists? (But that might be too anti-non-analytic philosophy to have the metaphysical meat you're looking for.) Kant? Can't know the history of mathematical thought without at least a brief stopover in Konigsberg. Penrose? Descartes is a mathematician. Leibniz. Leibniz. I rest my case. The Greeks. I rest my case again. Pierce. How about Godel's indispensibility argument? Maybe not, too many platonists in this list already. Three words: Alfred North Whitehead. And that's just scratching the surface. There's plenty of philsophy of mathematics to read. You just need to get that library card of yours out from the back of your wallet.

But that is my point. Their writings, the ones I know of anyway, are so mathematically flavored and use so much mathematical intuition that they can hardly be considered philosophy. Any math student after spending some time with certain mathematical concepts would come to the same conclusions.

How is that possible given that the philosophies are so divergent? How could a math student possibly conclude that Brouwer and Hilbert are both right? Edited November 14, 2005 by LauricAcid

[peoplater]

How is that possible given that the philosophies are so divergent? How could a math student possibly conclude that Brouwer and Hilbert are both right?

They both make valid points, one is a constructive mathematician and the other is in some sense a formalist, and both of their areas of research are now flourishing mathematical areas. And yes they are both right.

[LauricAcid]

They both make valid points, one is a constructive mathematician and the other is in some sense a formalist, and both of their areas of research are now flourishing mathematical areas. And yes they are both right.

I guess intuitionism is still vibrant, but it is still, as always, very much the minority view. A lot of work has been done since the foundational wars, by people like Gentzen, Kleene, and Godel, showing many of the important metamathematical relations between intuitionism and classical mathematics. It's an exciting subject.

[drewfactor]

Interesting discussion, I'm glad this thread has been revived. I am learning plenty since I am actually quite ignorant in math.

I agree with LauricAcid that many many of the great philosophers were also mathematicians. If they weren't considered mathematicians primarily, their ideas did have an impact on mathematics. Kantian constructivism is a perfect example. Descartes, Leibniz, and Spinoza all have had a major impact on calculus if I'm not mistaken. Russell and Whitehead have had a huge impact in the 20th century (although I can't say much for Russell's philosophy).

[peoplater]

So we all agree then. I am right and the only philosophy of math that makes any sense is the one that says mathematical concepts are just generalizations of everyday physical intuition. Everyone who says otherwise just doesn't know any better.

[Hal]

So we all agree then. I am right and the only philosophy of math that makes any sense is the one that says mathematical concepts are just generalizations of everyday physical intuition. Everyone who says otherwise just doesn't know any better.

It depends what you mean by 'generalizations of everyday physical intuition' - thats a pretty vague term which could mean almost anything. Its not obvious that things like ideals, Mandelbrot sets, or transfinite arithmetic have anything to do with physical intution though. The Mandelbrot set is an especially interesting case, since it probably constitutes the single strongest argument for Platonism I've encountered (Roger Penrose, a platonist, tends to use it a lot for this reason).

Edited November 16, 2005 by Hal

[Hal]

Of course I agree that people who don't know anything about mathematics are up to their necks when they pontificate about the subject. But, the vast amount of published work (I don't mean popularizations like the David Foster Wallace book and others) on the philosophy mathematics - journal articles, textbooks, et. al - is written by mathematicians or philsophers who have studied mathematics. Dedekind, Pierce, Frege, Hilbert, Russell, Brouwer, Heyting, Quine, Godel, Church, Carnap, Hintikka, Feferman, Putnam, Kripke, Dummet, Benacerraf and my favorite, Curry.

Well, I think there's a good argument to be made that too much modern 'philosophy of mathematics' has been written primarilly by logicians rather than mathematicians. I think that people will be far more likely to lean towards some kind of formalist/axiomatic viewpoint if they approach mathematics from the point of view of formal logic or philosophy, instead of actually learning the profound and deep mathematics which tends to lead to the prevalence of Platonism amongst practicing mathematicians. When someone is thinking about mathematics as 'all being reducible to arithmetic/set theory' rather than actually looking at the beauty which lies in certain results of advanced mathematics, its a lot easier to take nonsense like 'maths is just the study of formal systems' seriously.

Edited November 16, 2005 by Hal

[LauricAcid]

Well, I think there's a good argument to be made that too much modern 'philosophy of mathematics' has been written primarilly by logicians rather than mathematicians.

For example? Best would be an example of philosophy of mathematics written by someone actually remote from understanding even the importance of real analysis. Anyway, the fields of logic and mathematics intersect. Mathematical logic IS mathematics. One can't even approach discussing, for example, incompleteness without knowing such basics as the fundamental theorem of arithmetic and the Chinese remainder theorem. Or for example, one cannot even approach model theory without understanding number systems and algebraic structures.

I think that people will be far more likely to lean towards some kind of formalist/axiomatic viewpoint if they approach mathematics from the point of view of formal logic or philosophy, instead of actually learning the profound and deep mathematics which tends to lead to the prevalence of Platonism amongst practicing mathematicians.

Your argument is based on what you think is likely. But what IS contradicts you. Mathematician/logicians such as Godel and Putnam are decidedly in the platonist camp. And whether they are the minority, I don' know, but they are not vastly outnumbered. Platonism is quite common among logicians. On the other hand, the banner carriers for formalism are prominently mathematicians as well as logicians, as, of course, Hilbert, is the most salient example.

When someone is thinking about mathematics as 'all being reducible to arithmetic/set theory' rather than actually looking at the beauty which lies in certain results of advanced mathematics, its a lot easier to take nonsense like 'maths is just the study of formal systems' seriously.

But I don't know of any writer on mathematics who doesn't admire its beauty nor of any writer who contends that the study of formal systems precludes understanding mathematics in other ways. And the axiomatization of mathematics with set theory does not at all diminish one's appreciation of the beauty of the various branches of mathematics, but rather, adds to it. Edited November 16, 2005 by LauricAcid

[dondigitalia]

Well, you have to ask where the axioms originally come from. Normally a branch of mathematics is only axiomised once it is fairly mature - when you have a large number of results, people will start to see if there is some minimum number of basic statements they can all be deduced from, and these will become the standard axioms. The way that mathematics is presentd in textbooks isnt the way which it actually evolves - mathematicians do not generally start with a set of axioms and try to formally deduce things from them. Euclid's elements would be a classic example of this; most of the results contained in it were already known, although it was Euclid who first had the idea of reducing them to their basic premises. A more modern example would be something like group theory - the formalised notion of (eg) groups and ideals were postulated after many key results had already been found.

I agree; I'm not sure if this was intended as a disagreement with me or not, but I don't really see that this is inconsistent with what I said. The axioms are induced, but the laws, theorems, what-have-you that contribute to a complete mathematical theory are deduced from those axioms, which are then applied by induction.

You can also have disagreements about which axiom sets should be used - different axioms will have different consequences, so people can disagree about the best abstraction in a pariticular case.

That's true, too. People often disagree on applications of the same principles.

This is true psychologically, but not logically. Yes, its a psychological fact about humans that we learn the concept of the natural numbers before that of 0.

Well... that was kind of the point. That is how we, as humans, form the concepts. That is the fundamental starting point. That is the connection to reality, which is the subject that Don raised.

But this doesnt imply that the natural numbers come first from a mathematical point of view; the positive integers are normally defined in terms of the empty set with zero coming first: 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{}, {{}}} and so on.

Set theory is valid and useful in a great many contexts (and brilliant!), but it isn't a metaphysical base. It is an epistemological method.

Rational numbers do come after natural numbers in the standard constuction, since they are defined as equivalence classes of ordered pairs of integers, but there are different constructions where this is not true. For instance, when you construct your number system using something called surreal numbers, you start by creating a particular class of rational numbers (dyadic fractions) which you can then use to define the integers (and later, the real numbers).

Surreal numbers are far more abstract than either natural numbers or rational numbers, and are epistemologically much more complex; they are actually concepts of method. Using them in relation to real numbers is primarily a process of integration, not one of truly deriving the natural numbers from them. One cannot form the concept of "surreal number" without a long chain of abstraction leading back to natural numbers, which may be abstracted directly from reality, i.e. from the things we are using mathematics to measure.

[dondigitalia]

I disagree; I often use 'imagistic intuition' when solving math problems, and many mathematicians report the same. Sometimes the feeling of what the answer should look like occurs well before youve worked out how to actually reach it. There's a flash where you somehow 'see' the answer, even though you cant quite formulate the route which brings you there.

It is hard to describe, and in a sense, it's entirely proper to use the term intution. I've used the term myself regarding mathematics. I generally use "intution," in this context, to mean a subconscious suggestion deriving from automatized thought processes.

I do think that some strict Platonists regard it as "intuition" in the sense of direct access to knowledge by means of non-thought, but what they are actually experiencing is what I described.

Well thats certainly encouraging; I might order one of the backissues. The problem I've always found in the past with attempts to 'construct mathematics rationally' (or whatever) is that they tend to be exercises in psychologism, largely focusing on how humans acquire knowledge of mathematical concepts. And while that can be interesting from the point of view of cognitive/developmental psychology, it has very little to do with either mathematics or logic.

This view of mathematics has a great deal to do with mathematics and logic. It doesn't discount other studies, it strengthens them, by keeping them grounded in reality. The importance isn't necessarily one of theoretical mathematics, but one of applied mathematics. The lack of such a grounding is how we ended up with applications to non-reality such as all this hocus-pocus we have going on in physics right now (like in string theory).

It's the bridge between philosophy of science and foundations of mathematics (which, in my view, is a branch of the former).

[dondigitalia]

\You must mean we deduce our conclusions from axioms. The axioms are the premises; the theorems are the conclusions. But what do you mean by saying that we use induction to apply these? Do you mean in the technological application or do you have some application within mathematics in mind?

Axioms are axioms. They are the most fundamental premises. We arrive at the theorems by deducing from axioms, and then use those theorems as premises which we apply inductively. A proper theorem is universally valid; it is applied inductively, by acting as a single unit integrating all applications within a given context. While theorems may be applied inductively, we may also deduce further from them and arrive at various rules, laws, and formulas, all of which have an inductive purpose.

The point is that epistemologically, their purpose is primarily to integrate an innumerable quantity of individual applications into a single, manageable idea.

Specific applications... take your pick. L'Hospital's rule, the pythagorean theorem, the law of cosines... the list goes on and on.

[jrs]

A proper theorem is universally valid; it is applied inductively, by acting as a single unit integrating all applications within a given context.

I think that you are including both generalization and instantiation in your concept of "induction". Most of us consider instantiation to be deduction, not induction.

[dondigitalia]

I think that you are including both generalization and instantiation in your concept of "induction". Most of us consider instantiation to be deduction, not induction.

The act of applying the rule/law/theorem to an instantiation is inductive. In this context, I am using "applying" to mean: deciding that a specific problem is best solved using a particular rule/law/theorem, and initiating it's use. The instantiation which follows that induction almost always involves deduction, but oftentimes involves further inductions as well.

[Franklin]

First, I think I know what book you are reading. It is Brown's "The Philosophy of Mathematics: An Introduction to Picture and Proofs". This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

First, let's understand that Rand was not a scientist and that her ideas need to be understood in the context that gave birth to them. She once said that the amount of matter in the universe is fixed: This statement is patently false. Mass-energy is constant, not mass by itself. She clearly thought in Newtonian terms that have been shown inaccurate (I will not use the word false because they are TRUE IN MOST CONTEXTS. Rand herself writes about the context dependence of truth and the need to reformalize thought in the light of new information.

Mathematics is not the "science of measurement". This would completely eliminate number theory, the theory of polynomials, and much of analytic geometry from mathematics. Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies. It is foremost the study of identity---the internal consistency of the structures we use to discuss the outside world just as Grammar is the study of the way language is used to discuss the outside world. This analogy also holds who Math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in "The Principles of Mathematics Revisted"). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

First, I think I know what book you are reading. It is Brown's "The Philosophy of Mathematics: An Introduction to Picture and Proofs". This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

First, let's understand that Rand was not a scientist and that her ideas need to be understood in the context that gave birth to them. She once said that the amount of matter in the universe is fixed: This statement is patently false. Mass-energy is constant, not mass by itself. She clearly thought in Newtonian terms that have been shown inaccurate (I will not use the word false because they are TRUE IN MOST CONTEXTS). Rand herself writes about the context dependence of truth and the need to reformalize thought in the light of new information. She had little experience with mathematics, and her views represent the view of math that was dominant in the mid 1800's (probably the view held by her Professor Lossky). It is an incomplete view, esp. since the study of "measurement" entails many questions about the nature of space, number, set, function, and logic. To study "measurement" by itself would be to cut mathematics away from its basic context without reason or practical gain.

Mathematics is not the "science of measurement". This would completely eliminate number theory, the theory of polynomials, and much of analytic geometry from mathematics. Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies. It is foremost the study of identity---the internal consistency of the structures we use to discuss the outside world just as Grammar is the study of the way language is used to discuss the outside world. This analogy also shows how math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense. It is inspired by experience but relys equally on the observation of the structures of representation itself.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in "The Principles of Mathematics Revisted"). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

[LauricAcid]

Thanks for mentioning Hintikka, Franklin.

For other readers of this thread, I just mention that Hintikka, one of the giants of mathematical logic, issues a bold and fascinating challenge. Here are a couple of links:

Notice here not only the article by Hintikka and Sandu, but also Wang's article on Skolem and Godel, other articles about Skolem, and the article about first order logic:

http://www.hf.uio.no/filosofi/njpl/vol1no2/contents.html

Here's a review of Hintikka's book:

http://projecteuclid.org/Dienst/UI/1.0/Sum....rml/1081173780

Edited December 12, 2005 by LauricAcid

[dondigitalia]

First, I think I know what book you are reading. It is Brown's "The Philosophy of Mathematics: An Introduction to Picture and Proofs". This book has some definite virtues, but fails to be rigorous; (you should also not that his proof of Godel Incompleteness misuses some of the brackets.) He is very biased towards platonism.

Was this directed to me? You didn't quote anyone, and my post immediately preceded yours, so I can only guess. I've never heard of this book, so I can assure you, I'm not reading it.

Mathematics is not the "science of measurement"...Mathematics is the study of number, quantity, structure, space, change, and the logical systems that underpin those studies.

All of those things are either measurements themselves, or categories of things which must be measured. (What do you mean by "structure" in this context? The physical structure of existents, or the conceptual structure of a logical/mathematical system?) Numbers & quantity are both concepts of measurements. Space is a relationship between objects in temporal coexistence; change is the relationship between the characteristics of a single existent at different points in time. Both are things we measure.

This analogy also holds who Math is empirical while still not entirely relying on experience: It is exactly like grammar in this sense.

Well, yeah, the same can be said of just about any science. No knowledge is entirely empirical or entirely conceptual; there is always an element of each.

To put it simply, the best philosophy of mathematics is structuralist (mathematical entities gain their meaning from the structures they inhabit; this fights platonist trends), naturalist (inspired by observation of nature and the identity of representation itself) and weak constructivist (meaning like the construcivism of Jaako Hintikka; please read Constructivism Reconstructed in "The Principles of Mathematics Revisted"). I have discussed constructivism elsewhere here and what my problems with the stronger versions of constructivism are. Brown covers these ideas some too.

I'll thank you for mentioning Hintikka as well. I've had his work recommended to me a couple of times, but have never anything. I'm glad you reminded me.

[LauricAcid]

What measurement is involved in the theorem that there exists an empty set? What measurement is involved in, for example, the union axiom? In the theorem that Zorn's lemma is equivalent to the axiom of choice? In the theorem that every field of sets is a Boolean algebra? That every equivalence relation induces a partition? That there is always a finest partition? What measurement is involved in the definition of a lattice?

Edited December 12, 2005 by LauricAcid

[dondigitalia]

To say that mathematics is the science of measurement does not mean that every aspect of it can be reduced to a measurement. It is, in effect, an answer to the question: Why do we need mathematics? The answer: To measure things. It is the need to measure that gives rise to mathematics at all.

In regard to your specific questions, measurement isn't involved directly. But what's important to recognize is that, while extremely useful, set theory is not fundamental to mathematics as such. Ultimately, though, when applied to reality, mathematics is always measuring something.

[LauricAcid]

To say that mathematics is the science of measurement does not mean that every aspect of it can be reduced to a measurement. It is, in effect, an answer to the question: Why do we need mathematics? The answer: To measure things. It is the need to measure that gives rise to mathematics at all.

I think that's reasonable. But (1) I don't see that qualification given explicitly in, for example, OPAR, (2) Even if measurement is the primary motivation, I don't see how one can say that it is the only motivation, since so much of mathematics has to do with relations that actually omit measurement. Omission of measurement, for the purpose of greater generalization, is common in mathematics. The relations studied in mathematics are often not those that are about quantitative comparision.

But what's important to recognize is that, while extremely useful, set theory is not fundamental to mathematics as such.

There are other foundations proposed, but among your basic mathematics textbooks, you'd have to search high and low to find one that doesn't make use of sets, especially in such basic studies as analysis, topology, algebra, graph theory, and probability. One can choose all kinds of examples of mathematics that are about relations that are not quantitative from many areas of mathematics. I just chose those examples since they are simple ones.

Ultimately, though, when applied to reality, mathematics is always measuring something.

(1) That says something about application of mathematics more than it says about mathematics itself, (2) For example, studying Boolean algebras for the purpose of setting up, say, a switching design, is not quantitative. Example after example can be given. Edited December 12, 2005 by LauricAcid

[dondigitalia]

I think that's reasonable. But (1) I don't see that qualification given explicitly in, for example, OPAR, (2) Even if measurement is the primary motivation, I don't see how one can say that it is the only motivation, since so much of mathematics has to do with relations that actually omit measurement. Omission of measurement, for the purpose of greater generalization, is common in mathematics. The relations studied in mathematics are often not those that are about quantitative comparision.

Mesaurement-omission is key, but ask yourself what the ultimate goal is, in omitting measurements in mathematics. The assumption is that, at some point, you will put the omitted measurements back in again, for the purpose of arriving at some new measurement you were unable to make without mathematics.

There are other foundations proposed, but among your basic mathematics textbooks, you'd have to search high and low to find one that doesn't make use of sets, especially in such basic studies as analysis, topology, algebra, graph theory, and probability. One can choose all kinds of examples of mathematics that are about relations that are not quantitative from many areas of mathematics. I just chose those examples since they are simple ones.

I think you misunderstand me. I don't regard set theory as a foundation of mathematics; I think the foundation goes deeper than that, and is also more simple. But, this being the main focus of my own study in found. of math, I'm not willing to say much more than that right now.

(1) That says something about application of mathematics more than it says about mathematics itself, (2) For example, studying Boolean algebras for the purpose of setting up, say, a switching design, is not quantitative. Example after example can be given.

(1) It's a mistake to separate the function of any subject from the subject itself. (2)Who says all measurements are quantitative?

[Unconquered]

I think it would be helpful if you each defined the concept 'measurement'.

[dondigitalia]

I think it would be helpful if you each defined the concept 'measurement'.

Definitions are my Achilles heel, but off-hand the best I can give is: evaluation by comparison to an objective standard.