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[74.111.173.45]) by smtp.gmail.com with ESMTPSA id l5sm1866791qtu.42.2020.05.08.19.50.13 (version=TLS1_2 cipher=ECDHE-ECDSA-AES128-GCM-SHA256 bits=128/128); Fri, 08 May 2020 19:50:13 -0700 (PDT) Sender: Steven Awodey From: Steve Awodey Content-Type: text/plain; charset=utf-8 Mime-Version: 1.0 (Mac OS X Mail 11.5 \(3445.9.5\)) Subject: Re: [HoTT] Identity versus equality In-Reply-To: Date: Fri, 8 May 2020 22:50:12 -0400 Cc: =?utf-8?B?IkpveWFsLCBBbmRyw6ki?= Content-Transfer-Encoding: quoted-printable Message-Id: <87301D24-C9D1-4CD9-9A92-6287873BE131@gmail.com> References: <8C57894C7413F04A98DDF5629FEC90B1652F54CC@Pli.gst.uqam.ca> <1044A78D-8F16-4856-9B50-9B7FB7EF579A@gmail.com> To: Homotopy Type Theory X-Mailer: Apple Mail (2.3445.9.5) To elaborate a bit further in response to some private inquiries:=20 Brunerie=E2=80=99s proof that pi_4(S^3) =3D Z_2 , which is an entirely form= al proof within homotopy type theory, has two main parts: (1) it is shown that there is a natural number n such that pi_4(S^3) =3D Z_= n , (2) it is shown that n =3D 2. Since the proof of (1) is constructive, it produces an actual term t : Nat = . =20 In a system of type theory with the canonicity property=20 (like intensional MLTT w/o HITs or UA, or some of the more recent cubical t= ype theories), there is an algorithm which, from any such term t : Nat, will *compute* a n= umeral t* such that t* =3D=3D t (definitionally). This numeral t* denotes a natural number directly. =20 Applying the algorithm to (the term representing) =E2=80=9CBrunerie=E2=80= =99s number=E2=80=9D n in (1),=20 we should be able to use it to *compute* that n =3D 2, avoiding the need fo= r the (quite intricate) proof in part (2). =20 This is the part that (up to now) has not been possible to do *in practice*= , as Andre=E2=80=99 Joyal was pointing out. Thus we still require the proof in (2) to know that n =3D 2. The system of =E2=80=9CBook HoTT=E2=80=9D allows to formally prove many thi= ngs that cannot be computed in that system,=20 because terms coming from HITs or involving UA need not compute.=20 An advantage of cubical systems is that all terms compute =E2=80=94 in prin= ciple. But more practical work seems to be required in order to take advantage of = this feature in practice. Regards, Steve