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| In December 2006, David flew to Finland to give a lecture at Helsinki University on Friday 15th December and to be the opponent in the PhD examination of Markus Hähkiöniemi in Jyväskylä on Saturday 16th December. This involved a three and a half hour train journey from Helskinki to Jyväskylä via Tampere on Friday and back again on Sunday.
The thesis defence was a formal affair, with the candidate presenting his thesis first and his opponent asking him challenging questions. The major details have been accepted beforehand, so the whole thing is a civilised event, followed by a reception for the audience and then a celebratory dinner in the evening for academics, family and friends. It proved to be a magical experience. |
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 The audience at the talk on Proof and the role of Embodiment and Symbolism in Helsinki. |
 Anu Kähkönen, Antti Viholainen and Erkki Pehkonen. Erkki invited me to give the talk and Anu and Antti guided me on the train journy to Jyväskylä. |
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 Making a point. The mathematician in the front row is Juha Oikkonen who, after the talk, gave me his paper Mathematics between its two faces which is highly resonant with my formulation of the interplay between embodiment and symbolism. |
 ... Time for questions. |
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 The view from my hotel window in Jyväskylä on Saturday morning. |
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 David awaits the beginning of the PhD defence with Professor Pekka Koskela, who will chair the event. |
 Markus Hähkiöniemi presents his thesis, speaking in Finnish to a large audience of academics, family and friends. |
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 After his presentation, David, as the formal opponent, questions Markus about his thesis. |
 Markus responds to successive questions. |
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 The challenges continue. |
 Markus responds to one of his major ideas: a hypothetical path to the derivative that includes both embodiment and symbolism. His data challenges the APOS theory of Dubinksy and the operational-structural sequence of Sfard. |
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 David reads his formal report, declaring, “Any thesis which successfully challenges two major theories that have stood for a decade and a half deserves the highest accolade. I propose the thesis be approved with honour." |
 Professor Koskela bangs the gavil to declare the meeting over. |
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 The audience stream out of the lecture theatre ... |
 ... and go on to a celebration ... |
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 Congratulations! |
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 The department present Markus with a flower ... |
 ... and a vase to put it in (on the assumption that young mathematicians don’t have such things as vases for flowers). |
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In the evening there is a celebration dinner for the new Doctor of Philosophy, with his opponent as guest of honour. |
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 The two tables listen as the menu is explained ... |
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 After the dinner, Markus makes his speech thanking all those who have helped him, telling stories of his experiences. |
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 His opponent comments on his worthiness. |
 One of his supervisors, Professor Emerita, Maija Ahtee speaks as Markku Hannula and Markus’ partner, Tiina Kasi listen intently. |
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 Then Markku has his say ... |
 and it is open house as, one by one, others speak of their experiences of Markus. |
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 More presentations .... |
 ... these guys really know how to party! |
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