# What constitutes a definition and what is a mere property?

If a triangle is defined as a plane figure with 3 straight sides and 3 angles, would it be part of its definition that it has one less side than a square?

Or is that just a property of it?

How do I know which is part of the definition and which is a property?

Like, for example, the angles must add up to 180 degrees, is this part of the definition or is it a property? Also, is having 3 straight sides, which is part of the definition, also a property? I read the pages on definition and properties on plato.stanford, but they were no help. Thanking you in advance.

• Definition defines some properties of what is being defined, right? – user480281 Oct 15 '17 at 5:57
• Euclid defines a trilateral figure[I.Def.19] to be regions contained by three lines. He then goes on to define equilateral, isosceles, and scalene triangles[I.Def.20] as trilateral figures with one, two, or three distinct lengths for sides. Similarly obtuse and acute... The property that the angles add up to 180 degrees is not included in the definition (and indeed does not need to be true in non-euclidean geometry) and the relation to squares is absent as well. – JMoravitz Oct 15 '17 at 6:02
• Can I define a ball to be red? If i paint it blue, does it cease to be the same red ball? Or can red only be a property and not part of a definition? – Rob Hv Oct 15 '17 at 6:03
• So, it is up to the definer as to what is included in his definition? And you can put anything you want in your definition? As long as the definition is not contradictory? So I can define a blue monkey with a stapler for a nose and triangles for feet? – Rob Hv Oct 15 '17 at 6:04
• Technically, yes, the definer can include whatever he/she wants in a definition, however care must be taken to make sure that what you are defining is in fact "well defined." Also, if you are attempting to define an object that is already present in mathematical literature and lore, one should attempt to make sure that they define the same thing. It is generally considered elegant to have as little included in the definition as possible, letting all of the rest follow as corollaries or properties to be proven given the definition. – JMoravitz Oct 15 '17 at 6:09

How do I know which is part of the definition and which is a property?

You don't. A mathematician chooses the definition, and other properties then follow from that. A neat example is the determinant, which, as wikipedia notes, has various equivalent definitions. Not all mathematicians choose the same definition, although they do tend to choose equivalent definitions (I know of no exceptions).

This is also why you sometimes see comments on this site which ask the OP's definition of some concept: a question can be trivial to answer, or even true by definition, using one definition, but much more difficult using another equivalent definition.

• That is very helpful, thank you very much – Rob Hv Oct 15 '17 at 6:17

Like Jmoravitz says, whatever you are defining, it should be well defined. Once defined, there should be no ambiguity about what object would be. Example: A definition of a square is that it is a quadrilateral with all equal sides and all equal angles. A property that follows is that the sum of a pair of opposing angles is 180 degrees. Now Lets change "property" into "definition": A definition of a square is that a pair of opposing angles is 180 degrees. Is this a well defined definition of a square? Does that mean we are talking about a square? Could be, or not. After all, a cyclic quadrilateral also has pairs of opposing angles summing 180 degrees. Cyclic quadrilaterals often are not squares. So using a definition to introduce a particular entity should be subject to a more stringent "test of correctness" than a introducing this entity by revealing one of its properties

Accidental properties and essential properties is what I was after.

An essential property is one that an entity, without, would not be itself. An accidental property is one which, without, it would still be itself.