Brouwer’s Weak Counterexamples and the Creative Subject: A Critical Survey

I survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering (i) what useful mathematical work is done by weak counterexamples; (ii) whether they are rigorous mathematical proofs or just plausibility arguments; (iii) the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; (iv) what axioms for the creative subject are needed; (v) what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with a weak counterexample of my own. I also examine Brouwer’s 1927 proof of the negative continuity theorem, which appears to be a weak counterexample reliant on both the creative subject and the concept of choice sequence; I argue that it provides a good justification for the weak continuity principle, but it is not a weak counterexample and it does not depend essentially on the creative subject.

Neural activity in human ventromedial prefrontal cortex reflecting the intention to save reward

Saving behavior usually requires individuals to perform several consecutive choices before collecting the final reward. The overt behavior is preceded by an intention to perform an appropriate choice sequence. We studied saving sequences for which each participant rated the intention numerically as willingness to save. Each sequence resulted in a specific reward amount and thus had a particular value for the participant, which we assessed with a Becker-DeGroot-Marschak auction-like mechanism. Using functional MRI, we found that blood-oxygen-level-dependent signals in human ventromedial prefrontal cortex (vmPFC) correlated with the participant’s stated intention before each choice sequence. An adjacent vmPFC region showed graded activation that reflected the value of the sequence. These results demonstrate an involvement of vmPFC in intentional processes preceding sequential economic choices.

On winning shifts of marked uniform substitutions

The second author introduced with I. Törmä a two-player word-building game [Fund. Inform. 132 (2014) 131–152]. The game has a predetermined (possibly finite) choice sequence α1, α2, … of integers such that on round n the player A chooses a subset Sn of size αn of some fixed finite alphabet and the player B picks a letter from the set Sn. The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue–Morse substitutions. It is known that W(X) and X have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue–Morse substitutions.

The Differential Outcomes Effect Using Sensory Outcomes in a Many-to-One Matching-to-Sample Task

The present study was designed to evaluate the use of sensory outcomes (visual vs. auditory) using a differential outcomes procedure to facilitate learning in a many-to-one matching-to-sample task. For one group of participants (differential outcomes) each correct stimulus-choice sequence was always followed by a different outcome; whereas for the rest of participants (non-differential outcomes) each correct sequence was followed by the same outcome. Participants trained with differential outcomes showed a faster acquisition and higher overall accuracy than participants trained with non-differential outcomes. The results provide a new extension the differential outcomes effect by using sensory outcomes and many-to-one matching to-sample task; applications of the differential outcomes procedure are discussed.

INTUITIONISTIC ANALYSIS AT THE END OF TIME

Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. “Not every sequence is pointwise equal to a definite sequence” is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left( {\omega ,{\cal C}} \right)$ of C°.

Primate amygdala neurons evaluate the progress of self-defined economic choice sequences

The amygdala is a prime valuation structure yet its functions in advanced behaviors are poorly understood. We tested whether individual amygdala neurons encode a critical requirement for goal-directed behavior: the evaluation of progress during sequential choices. As monkeys progressed through choice sequences toward rewards, amygdala neurons showed phasic, gradually increasing responses over successive choice steps. These responses occurred in the absence of external progress cues or motor preplanning. They were often specific to self-defined sequences, typically disappearing during instructed control sequences with similar reward expectation. Their build-up rate reflected prospectively the forthcoming choice sequence, suggesting adaptation to an internal plan. Population decoding demonstrated a high-accuracy progress code. These findings indicate that amygdala neurons evaluate the progress of planned, self-defined behavioral sequences. Such progress signals seem essential for aligning stepwise choices with internal plans. Their presence in amygdala neurons may inform understanding of human conditions with amygdala dysfunction and deregulated reward pursuit.