Posted on April 26, 2015 by

A while back I was reading a paper  by Bob Hale and Crispin Wright, first published in 2000 [1], in which they use the symbol “#” to denote an arbitrary matrix sentence. That was over half a decade before twitter went live. That symbol, variously known as the hash sign/symbol, the number sign, or octothorpe (my favourite, though George Boolos is the only philosopher I’ve seen use that term), has a number of distinct uses in philosophy, but all of the philosophical uses combined are completely overshadowed by the recent rise of hashtags in social media.

The uses I am aware of in philosophy are:

  • the use mentioned above, as a place holder for an arbitrary (truth-evaluable) matrix sentence;
  • as a function symbol or term-forming operator for cardinal numbers used mostly by neo-logicist/abstractionist philosophers of mathematics; and
  • in linguistics and philosophy of language, as an indicator of badness, i.e. # There are four water over there.

I imagine there are others, and would be interested if readers were to point them out. In any case, when reading the Hale and Wright piece, I found myself reading the construction “#f” as “hashtag-eff” due to the pervasiveness of hashtag terminology in the media, and realized people familiar with the octothorpe from different contexts might read it in very different ways leading to some possibly amusing results.

Apart from its ubiquitous presence on telephones, most of my experience with the octothorpe is its use as a term forming operator for cardinal numbers (the second bullet, above), usually read as “the number of…”. This comes from the rendering in modern notation of Hume’s Principle (HP) along the lines of #F=#G↔F≈G, read as “the number of the Fs is equal to the number of the Gs if, and only if, the Fs and Gs are equinumerous (i.e. there is a bijection between them)”.

Not surprisingly, #GameofThrones is trending on twitter. On the most natural reading for me, with sensible syntax anyway, is “the number belonging to the concept ‘GameofThrones'”, which is presumably ‘1’ (or maybe 2, if the book and the series both fall under the concept). Given that reading, presumably the referents of #GameofThrones and #BarackObama are the same–the number one. That said, I do think it is an interesting question whether hashtags have referents.

Given the popularity of the program, the third reading above is even more amusing: “Game of Thrones –bad (liguistically)”. The first reading seems perfectly fine–“there’s a possible truth-evaluable sentence containing the term ‘Game of Thrones”–but it would still be odd to read Tweets in that way.

In the other direction, people for whom the use of hashtags preceded their reading of philosophy of mathematics and language may naturally give readings of our above examples (out-loud, if you like) as something like “hashtag There_are_four_water_over_there.” and “Hashtag F is equal to hashtag G iff…”, or maybe, given that ‘F’ and ‘G’ are variables, “The hastag of F is equal to the hashtag of G…” This is less amusing than the reverse cases, but certainly a possible source of confusion.

The phenomenon highlighted here is, I think, part of a larger issue with the extreme context sensitivity of many terms and symbols used by philosophers. For example, if I were to tell you that I’m an intuitionist (I’m not, by the way), I could either hold a fairly popular view in (meta-) ethics, or an extremely unpopular view about mathematics and logic. In the symbolic case, “→” could be an embedding (in set-theory), a function, an arrow (in category theory or group theory), a material conditional, or a non-material conditional, depending on the context.

The case of the octothorpe is particularly interesting because of its prevalence outside of philosophy and mathematics.

[1] Hale, Bob and Crispin Wright, “Implicit Definition and the A Priori”. In The Reason’s Proper Study, Oxford: OUP, 2001.