We will hold our first meeting of the semester, however, this coming TUESDAY (September 27th) from 5-7pm in TB130. In this meeting we will decide what to read in the coming semester (so, please come with suggestions). And, we will discuss the following article:
George, Rolf (1981). Kants Sensationism. Synthese 47 (2):229 – 255.
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This reading group is part of the joint Boğaziçi -Southampton Newton-Katip Çelebi project AF140071 “Agency and Autonomy: Kant and the Normative Foundations of Republican Self-Government” run by Lucas Thorpe (Boğaziçi) and Sasha Mudd (Southampton) and by Lucas Thorpe’s Bogazici University BAP project 9320
Our philosophy/cog-sci reading group will continue this semester on Mondays from 5-7pm in TB365. The topic for the start of the semester will be affordances. Everyone is welcome.
We will begin this coming Monday (26/09/2016) by discussing the following article:
Lorenzo Jamone, Emre Ugur, Angelo Cangelosi, Luciano Fadiga, Alexandre Bernardino, Justus Piater and Jose Santos-Victor, (2016) “Affordances in psychology, neuroscience and robotics: a survey” IEEE TRANSACTIONS ON COGNITIVE AND DEVELOPMENTAL SYSTEMS, VOL. XX, NO. XX, XXXX.
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This reading group is organised as part of Lucas Thorpe‘s TÜBİTAK project “Concepts and Beliefs: From Perception to Action” ( 114K348).
Thursday, September 22nd, 2016 (3pm -5pm, TB130):
Friday, September 23rd, 2016 (4pm-6pm, Venue TB130):
Cantor gave two proofs that the cardinality of the set of real numbers is greater than that of the set of natural numbers. In the better known of these proofs, his diagonal argument, Cantor randomly tabulates the real numbers in the interval [0, 1) in a square array. He asks us to assume for reductio purposes that this table is “complete” in the sense that it contains the totality of real numbers in that interval. He then constructs a new number from the diagonal of the table. He asserts that this constructed number is a new real number in [0, 1) which cannot be found in the putatively complete list. His famous conclusion is that no table that lists real numbers in that fashion will have the items in it being enumerable by natural numbers. Since no such list will manage to include all of the real numbers in [0, 1), it follows, according to Cantor, that there have to be more real numbers than natural numbers. Hence the size, or cardinality, of the set of real numbers is greater than that of natural numbers.
I will use a binary version of Cantor’s table for ease of exposition:
d = 0.11000110100…
i = 0.00111001011…
Thus we begin with the Cantorian assumption that this infinitely long and random list includes all the real numbers in the interval [0, 1). The numbers on the diagonal of the list (boldfaced in the table) give us the infinitely long number d. From d, Cantor would obtain the number i, by substituting every 0 after the decimal point in d by 1, and every 1 in d by 0. Now, the number i couldn’t be found in the list, according to Cantor, because i is guaranteed to differ from each binary number in the list since the number’s digit that falls on the diagonal is changed. Therefore, no list of real numbers (in this example the reals in [0, 1)) can be enumerated by natural numbers, which entails that there are more real numbers than natural numbers.
The problem I see with Cantor’s argument is the following. I will call two binary numbers in [0, 1) “inverses” of each other if one number has 1 in a certain location after the decimal point, the other number has 0 in the corresponding location; and if it has 0 in that location, the other number has 1. Thus the numbers on lines 3 and 7, for instance, are inverses of each other. So are the numbers on lines 5 and 10. Crucially, d and i are also inverses of each other. If we are to assume this list is complete, every number in the list must have its inverse also included in the list. Cantor says i is not included in the list. But how come? The number i’s inverse, namely d, is in the list—it is on line 8. Since its inverse is in there i must also be in there, like every other pair of inverses.[*] Suppose d’s inverse was in the list, say it was on line 17. Cantor would say his diagonal trick would change the 17th digit of the number on line 17, and therefore i could not be in the list.
So we have a contradiction: (a) The number i has to be in the list, say in line 17, because its inverse d is. (This is my claim, not Cantor’s.) (b) Yet i cannot be in the list, because it differs from the number on the 17th line in its 17th digit. (This is Cantor’s claim.) If one thinks (a) and (b) both sound plausible, as they might, one would find himself facing this paradox:
(i) We can make a random list of real numbers in [0, 1).
(ii) This list can be assumed to be exhaustive.
(iii) The diagonal d of the list and i can be fully formed.
(iv) The number i has to be in the list, if the list is to be exhaustive, since its inverse is.
(v) The number i cannot be in the list, since it is different from every number in the list in some n-th digit.
Since (iv) is impeccable, Cantor’s diagonal procedure which derives (v) can be faulted with leading to paradox, and consequently his denial of (ii) can be claimed unjustified. Hence his entire diagonalization procedure must be flawed. Therefore his claim that there are more real numbers than natural numbers is unwarranted.
As for the flaw in Cantor’s diagonalization procedure, I think it arises from the following phenomenon. The diagonal and its inverse clash at some digit. Suppose the 17th digit of the diagonal is 1; then the 17th digit of its inverse would have to be 0. But, since the two digits overlap, the diagonal’s 17th digit being 1 rules out the would-be inverse’s 17th digit being 0. Thus the diagonal disallows its inverse (with the correct 17th digit 0) from being a member of the table.
Hence, we need not get impressed when Cantor comes up with a number i which is not in the table. We know that i would have been in the table if it had not been censored by the very diagonal of the table.
It should now be obvious that the problem of missing i arises from the fact that Cantor lists real numbers in the form of a table. That turns out to be a deceptively misleading way of displaying real numbers.[**]
[*] If i were not to be in the list although its inverse is, we could not assume the list was complete to begin with.
[**] My more detailed criticisms of Cantor’s arguments and my way of showing that reals cannot be more numerous than naturals can be found at: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_
Talk at Bogazici, Jonathan Cohen (UCSD), “On the Presuppositional Behavior of Coherence-Driven Pragmatic Enrichments”
Please join us for this talk.
Consider (1) and (2):
(1) Every time the company fires an employee who comes in
late, a union complaint is lodged.
(2) If the company fires an employee who comes in late, a
union complaint will be lodged.
Now suppose there is an employee, Snodgrass, who is fired because he
was discovered to have been embezzling, and that the firing occurred
on a day on which he had happened to come in late. And suppose the
union does nothing about it. In this situation, it seems that (1) and
(2) can still be true.If so, there must be an enrichment at play that affects truth
conditions: even though Snodgrass came in late and was fired, the fact
that Snodgrass wasn’t fired *because* he was late causes the
event to sit outside of the domain restriction of (1), and, likewise,
not to satisfy the antecedent of the conditional in (2). Importantly,
this enrichment has no linguistic mandate, and therefore is clearly
pragmatic. So why does it intrude upon the truth conditions of (1) and
We argue that this behavior arises as a result of the way in which the
enrichments interact with pragmatic presupposition. In particular, in
producing examples (1)-(2) with the intention to communicate the
enrichment, a speaker presupposes that a causal relation necessary to
make the inference is part of the common ground, and intends that
presupposition to restrict the interpretation of the
quantifier/antecedent in (1)/(2).
The analysis connects the associative inferences that underlie the
establishment of coherence relations between sentences in a discourse,
a class of intrasentential enrichments that result from the same
principles, and the manner in which presuppositions constrain the
interpretation of quantified expressions and conditionals.
Applications for Masters in Philosophy (Round 2)
We are now accepting the second round of applications for the M.A. in Philosophy (for those starting the degree in Fall 2016).
Up to five successful applicants will have the opportunity to spend a semester at Australian National University. In addition successful applicants will receive a comprehensive scholarship (tuition waiver, accommodation and monthly stipend including private health insurance).
This is a two year masters that includes coursework and a thesis. The language of instruction for all courses is English.
Admission requirements and online application can be found here.
Philosophy and non-philosophy majors are encouraged to apply. We also warmly welcome applications from international students.
Application deadline (Round 2): 18 July 2016
Written exam (Round 2): 25 July 2016**
Interview (Round 2): 27 July 2016**
*Those planning to take the COPE exam on June 28-30 should contact us at the above email address by June 22 to reserve a place.
** International students may take the written exam remotely and complete the interview via skype.