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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.

  • According to Kant, angles of a triangle need not add upto 180 degrees per a definition of triangle. – Mozibur Ullah Oct 15 '17 at 5:17
  • @MoziburUllah really? why ?? – user28660 Oct 15 '17 at 5:50
  • @user3293056: It's in his book The Critique of Pure Reason; I've been pointing this out to people here and there because Kant is usually taken to be against non-Euclidean geometry; whereas on this account it can be plausibly suggested he helped open up this route; his main aim there was to distinguish what is empirically true and what we can deduce from definitions - and he was saying merely from the definitions it wasn't the case the angles need add upto 180 degrees – Mozibur Ullah Oct 15 '17 at 5:59
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    You need to specify which school of philosophy you're asking about ... if this is about how philosophers understand definitions. – virmaior Oct 15 '17 at 13:57
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    There is no such thing as "the definition" of a thing. You can choose whatever list of properties specifies a thing uniquely (so-called definite description), and make that its definition. If you wish the definition to be more "substantive" you can additionally require that the description only include essential properties. But even then there are multiple options. If triangle is defined as in the OP then nothing else will be part of the definition, although it may follow from it. – Conifold Oct 15 '17 at 22:51
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What properties constitute a definition? It depends on whom you are raising the question to, as virmaior commented. The SEP article you mention offers some families of definitions of 'definition' as well. But if we confine ourselves to matters relating to definitions in a formal system, we should be talking about the debate between Frege and Hilbert at the turn of 20th century. The two intellectual giants fought over what ‘definition’ must mean. The memorable quote for beer loving logicians and mathematicians is a by-product of their debate:

One must be able to say at all times -- instead of points, straight lines, and planes -- tables, chairs, and beer mugs.

A lesson from several notorious pseudo proofs of 19th century led Hilbert to hold the view that all proofs must be devoid of the tyranny of intuition. In his “The Foundations of Geometry”(1899), Hilbert set out to redo Euclid’s geometry (from synthetic geometry to analytic geometry). Hilbert thought that interpreting primary terms like point, line and plane would introduce the so-called unwanted spatial intuitions that would lead us astray. Hilbert maintained that the foundations of geometry should be given without defining point, line and plane. According to him, we know how to use these words because their meanings are implicit in axioms which specify relations among these terms. Understanding in this way, Hilbert interpreted a point as a real number in coordinates, and talked about theorems of real numbers to talk about geometric sentences.

Frege sent a letter to Hilbert to complain that, in Hilbert's system, "the axioms are made to carry a burden that belongs to definitions. To me this seems to obliterate the dividing line between definitions and axioms in a dubious manner,... " (Gottlob Frege, Letter to Hilbert of 27 December 1899). Frege believed that a mathematical system must begin with clear definitions of terms and then proceed to axioms(true but unprovable statements) and then theorems(true and provable statements). To Frege, symbols and formula have meanings (i.e., senses and thoughts). So to Frege, geometry must begin with defining point, line and plane. For this reason, Frege seemed to believe that the talk of real numbers to talk about geometry amounted to a fallacy of weak analogy.

Due to their differences in what constitutes a definition, “while Frege takes it that Hilbert owes an explanation of the inference from the consistency of AXR to that of AXG, for Hilbert there is simply no inference.” (https://plato.stanford.edu/entries/frege-hilbert/. Here, AXR refers to an axiomatic system of real numbers and AXG refers to an axiomatic system of geometry) In response to Frege’s demand of explicit definitions for primary terms, Hilbert famously responded with the above memorable quote.

  • If we confine ourselves to definitions in a formal system what difference does it make how one interprets its axioms, according to Hilbert or to Frege? Triangles can be defined by the same collections of properties whether one treats point and line as interpreted or uninterpreted symbols. – Conifold Oct 17 '17 at 0:33
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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?

If a triangle (see Euclid, Def.19) is defined as a plane figure "which is contained by three straight lines" and a quadrilateral one that is contained by four, then it is a consequence of the two definitions that the triangle has one side less than a square.

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