Show, Prove or Demonstrate, that is the question

In mathematics textbooks and class notes (especially from undergraduates onwards), we often see these three verbs: show, prove, demonstrate.

So we can ask ourselves, what are their meanings?

First, let’s look at the meaning of these words in the language. For this, we performed a search on the website https://www.dicio.com.br/ and arrived at the following results:

Show:

  • Display, make see; expose to eyes give evidence of; make explicit, visible; to demonstrate
  • Express yourself privately or intimately; reveal or reveal
  • To indicate or indicate through gestures, signs or traces; aim appear

Prove:

  • Demonstrate the truth, reality, authenticity of a thing with reasons, facts, testimonies, documents, etc. bear witness to; Show; to demonstrate; submit to the test; give proof of
  • Know from personal experience; to experiment; suffer, suffer
  • Put on before you’re ready to see if it looks good
  • Eat or drink to know if it’s good

Demonstrate:

  • To make (something) evident through evidence; make it known; prove
  • Exposing (feelings) through the use of external signs; express
  • View features, attributes. qualities etc.
  • Demonstrate what you intend to explain or what you are explaining; exemplify
  • Express particularities of your own personality; reveal yourself

In the meanings obtained, we highlight those that best resemble the way these verbs appear in Mathematics textbooks and class notes. Note that the confusion of these terms even resides in the linguistic context, as we highlighted underlined, where each verb uses the other two verbs in its explanation.

However, the multiplicity of meanings that generate confusion does not seem to reside in a specific verb. We can come across a book or article that claims to show, prove or demonstrate something in Mathematics, and when we see it, there is a visual scheme, a sketch of the idea, an indication of the way of thinking that leads us to “believe” that that property it’s true. In the context of Basic Education, the situation is a little more complicated, as these verbs are often associated with examples.

All this confusion of meanings changes when the words Mathematical formalism or Mathematical rigor appear:

  • Show/Prove/Demonstrate with mathematical formalism…
  • Show/Prove/Demonstrate with mathematical rigor…

This is a clear sign that examples, visual schemes, drafts of ideas or reasoning will not be enough to meet what is asked.

We have arrived at the hard part of Mathematics, in which we need to present arguments that justify for some domain, which determines property is valid or invalid. In this case, when We Show/Prove/Demonstrate something with mathematical formalism, we usually refer to this argumentative text as Proof or Demonstration. For example:

Show/Prove/Show with mathematical formalism that “every odd number squared is odd”.

  • Proof/Demonstration:
    • Let x be any odd number.
    • So x can be written in the form 2.n+1 where n is an integer.
    • So x² = (2.n+1)² = 4.n² + 4n + 1.
    • But 4.n² + 4n + 1 can be written as 2.(2.n²+2n)+1.
    • Since (2.n²+2n) is an Integer, so 2.(2.n²+2n)+1 is odd.
    • Therefore, every odd number squared is odd.

A subtle difference however occurs when the claimed statement is false. In this case it is strange (although nothing forbids it) that we say to Show something that is false. While it seems more natural to use the terms Prove/Demonstrate in connection with something that is false.

Prove/Prove with mathematical formalism that “every prime number squared is odd”.

  • Proof/Demonstration:
    • The statement is false.
    • Take the prime number 2.
    • 2² = 4, but 4 is not odd.

In practice outside the walls of mathematical formalism there is no restriction on the use of these terms. Even George Pólya advocates that we omit or disguise some of the more abstract and complex steps of a proof, to enable its presentation to audiences that still do not have the level of mathematical sophistication necessary to fully understand it (Pólya calls them informal proofs).

But here’s a tip, if you’re enrolled in a Mathematics discipline since graduation, or you’re reading something aimed at the university audience of Mathematics, always understand these three verbs as synonyms. When questions about show/prove/demonstrate arise, try to recognize the domain of that property and find arguments that guarantee its validity or invalidity.

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