# Can you Define Something in Terms of What is True About it?

Is it possible to define something in terms of what is true about it?

Suppose for example, the case where in mathematics a definition is given in terms of formal symbols and rules for which strings involving them are designated as 'true'.

• By the way, Dean, off-topic but, noticing your uploaded image, see the voronoi diagram on my homepage, or just click forkosh.com/images/avoronoi.gif (my voronoi's bigger than your voronoi:) – John Forkosh Aug 21 '17 at 8:38
• Very nice! I also like my veronoi because it shows that irregular pentagons can tesselate the plane. – Dean Young Aug 21 '17 at 8:53
• What I found even more interesting than irregular pentagons is the tesselation including pentagons, pentagon-like shapes with straight-edge and curved sides, up to and including perfect circles. That automatically emerges from all-straight-lines when you assign weights to the distances, i.e., the Euclidean distance from an arbitrary point to each object is weighted (typically linearly weighted) by an object-assigned weight. And then the nearest object is determined among those weighted distances. Flip around those weights and the whole thing dramatically changes. – John Forkosh Aug 21 '17 at 22:55

## 2 Answers

Physical systems, more precisely the state of a physical system, is also defined "in terms of what is true about it". In this case, the "true statements" are yes/no propositions that can be operationally tested by constructible apparatus (very much a logical-positivist-like approach, but arising from strictly physical considerations).

Then, measuring the yes/no-outcomes for a maximal set of consistent propositions completely determines the measured system's state. Here, "consistent" means simultaneously measurable, i.e., no measurement disturbs the already-determined outcome of any other. And "maximal" means that any additional measurement necessarily distrurbs one or more already-determined outcomes.

And, in fact, there's no other way to define a physical system's state. So, not only "can you define...", but you've got no choice "but to define...".

Edit...
...
too-long-for-a-comment answer to @Gordon's comment below

The remarks above are just a standard result. It's typically derived as a quantum result since classical propositions are always consistent (also called "compatible"). But the same idea holds for everyday measurements of macroscopic systems, as follows.

Firstly, as I parenthetically remarked above vis-a-vis logical positivism, particularly the verifiability theory of meaning, a proposition is associated with the procedure or apparatus used to verify/measure it. And then the general idea here is that whenever two or more propositions are compatible, you can construct a single apparatus that simultaneously measures them both (or them all).

You can see that exactly stated in https://books.google.com/books?id=dVs8PcZ0Hd8C&pg=PA115 (just above the commutator, Eq.6.25). I'm not sure why Isham calls that "trivially compatible" (I'm not sure how they'd commute otherwise). So let's just overlook that wrinkle (unless somebody can followup with a counterexample).

The ultimate upshot of this is that any observable can be decomposed into a maximal set of yes/no-propositions (which are just observables with exactly two possible outcomes/eigenvalues). And that's precisely stated and derived somewhere in Chapter 5 or Chapter 6 of https://books.google.com/books/about/Foundations_of_quantum_mechanics.html?id=FwpRAAAAMAAJ But I'm not immediately finding it in my printed copy, which I studied carefully some years (or decades) ago. And google seems reluctant to display entire pages, anyway (though I did notice an easy-to-find pirated pdf you can download, but won't explicitly give out the url).

Perhaps the easiest-to-follow discussion comprises the first six pages (right, just six) of https://books.google.com/books?id=WWYbAQAAIAAJ at which point Schwinger concludes "...the symbol of this compound measurement is...[math elided]...which then describes a complete measurement, such that the system possesses definite values for the maximum number of attributes; any attempt to determine the value of still another independent physical quantity will produce uncontrollable changes in one or more of the previously assigned values." (he didn't seem to have much use for periods)

Anyway, the above quote's from pages 5-6 (and you might likely want to read through page 12), but google also seems reluctant to display entire pages of this book, and I'm not seeing any pdf's. Schwinger's subsequent https://books.google.com/books?id=fDX6CAAAQBAJ&pg=PA001 develops that discussion much more formally, and is a correspondingly harder read, but google seems willing to display pages.

Finally, re Boltzmann, no. These are complete measurements determining pure states, not density matrices determining canonical ensembles. That wouldn't really address the OP's question regarding "things" (I'm taking his "something" to explicitly mean thing rather than ensemble). Whatever philosophical argument you want to have about the definition of "thing", a pure state is as close to "thing" as you're going to physically get.

• What branch of knowledge does this fall under? I have heard this before, is this a general principle in science? Is this a type of measurement that could only be done through probabilty like Boltzmann? I'm trying to figure out how one would make such a measurement. – Gordon Aug 20 '17 at 11:58
• @Gordon see Edit in answer for reply to your comment. – John Forkosh Aug 21 '17 at 7:31
• John this is an excellent response to my question. I appreciate the time you spent on it very much. I have very little math and science background so I try to learn as I go along. – Gordon Aug 21 '17 at 22:40

You give the example yourself: Yes, algebraic structures, such as groups, rings, fields, or lattices are defined in exactly this way. The definition of, say, a group does not specify what the set of elements and the operations of a group are, but it specifies what kind of formulas these elements and operations must necessarily satisfy – not more, and not less.

• Beat me to it! It's possibly also worth mentioning that defining objects in this fashion makes them versatile -- if we subsequently happen to find some other 'concrete' object which exhibits the 'true' properties of our abstractly defined object, we immediately know many (presumably useful) things about this new concrete object. – Alec Rhea Aug 20 '17 at 3:28
• Or the definition may be made in terms of what is false about it. The 'via negativa' and all that. – PeterJ Aug 20 '17 at 11:23
• @AlecRhea interesting. I think I could use this as an example of idealism. The rock itself disappears into abstraction! No doubt useful however. – Gordon Aug 20 '17 at 14:36