In mathematics one is often presented with a theorem, followed by a proof. At the end of the proof, one encounters commonly two phrases:

"We conclude the proof of the theorem..."


"We thus conclude the theorem".

Does this second sentence use the word 'conclude' properly? I believe it does, though it is obviously in a slightly different sense than the first. The first sentence says "We end the proof", while the second says "We judge the theorem to be true".

I would appreciate a clarification of this.


Conclude has two senses which might interest you:

1. [WITH OBJECT] Bring (something) to an end:

'they conclude their study with these words'


2. Arrive at a judgment or opinion by reasoning:

'the doctors concluded that Esther had suffered a stroke'

'what do you conclude from all this?'

"We conclude the proof" is an example of sense (1).

"We thus conclude the theorem" is an example of sense (2). Syntactically and semantically there is nothing wrong with it. But it does sound a bit funny, probably because of the ambiguity of "conclude". There are much better ways of expressing the same thing. For example:

  • Thus, the theorem is true.
  • Thus, the theorem is proved.
  • We have thus proved the theorem.
  • We have thus established the theorem.

That said, a Google search reveals that many people do use "conclude the theorem", despite the fact that it sounds a bit funny. They often use it synonymously with "prove the theorem." For example:

  • Yes, I've seen the google searches and that was the basis of my question. My only comment is that the second definition (which I've seen) uses in the example the "that" clause as the object of what is concluded. That was the more specific confusion I had. Should it be "We conclude that the proof is correct"? – Lepidopterist Aug 19 '16 at 2:34
  • You can use a that clause but it's not necessary. You can, for example, say "We conclude Goldbach's Conjecture from corollary 1 above." No that clause is necessary. – GoldenGremlin Aug 19 '16 at 12:36

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