Logic programming has been introduced as programming in the Horn clause subset of first order log... more Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a non-monotonic reasoning formalism strongly related to Default Logic and Auto-epistemic Logic.
It is proposed that the decision making process is intrinsically formulative in nature; i.e., for... more It is proposed that the decision making process is intrinsically formulative in nature; i.e., for the individual the crux of the process is in the development of a frame, a knowledge base, that can be used to make the decision, rather than the evaluation of the alternatives ’per se’. A two stage model of decision making is forwarded. In the first stage the individual develops his frame. This is the critical, formulative, stage and dependent on the socio‐cultural environment. In the second stage the individual makes a decision by evaluating his frame. This stage is mechanical, evaluative, and determined by the frame formulated during the first stage. A consequence of the basic thesis is that the social and cognitive processes can not be decoupled if we are to understand decision making behavior. However, there are few methods currently available that allow the researcher to look at the relationship between the social and the cognitive process.Herein, a set of methods that permit the researcher to look at this relationship are described and then applied to a particular case study. On the social side these methods are based on a network representation of social structure. On the cognitive side these methods are based on a set of semantic models of human knowledge. A set of cognitive models for representing information are presented — these include frames, maps, and knowledge bases. A set of programs, Frame Technology, for coding, manipulating, and analyzing frames, based on these models, are briefly described. The relationship between social structure and individual cognitive formulations for a group of undergraduates at MIT formulating the problem of selecting a new tutor is then examined using these programs.This work is an overview of a new approach to studying decision making, one that tries to relate social structure to cognitive structure at an empirical level. It is not a detailed analysis of a particular decision: the point is not what decision was made, but the role that information played in this decision. Nor is a detailed discussion of the set of new methods provided: the point is not how the methods work, but the way in which they can be used.
Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of hist... more Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have ... more In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be extended to tensors. The concept of rank is itself difficult to define, and its calculation raises difficulties. Numerical algorithms have nevertheless been developed, and some are reported here, but their limitations are emphasized. These reports hopefully open research perspectives for enterprising readers.
Evolutionary psychology can shed light on mathematical abilities. To do so, it must be supplied w... more Evolutionary psychology can shed light on mathematical abilities. To do so, it must be supplied with a strong model of cultural evolution. Natural selection endowed us with some modules, like the number sense or the logical module, useful for mathematics. Then cultural evolution tinkered with them to create modern mathematics. Of particular significance during cultural evolution is ancient Greece, where a feedback loop that stressed the importance of formal proofs was engaged. Ever since, mathematicians have been engaged in a quest for relevance that uses formal proofs as an indicator of relevance. This led to modern mathematics. Nonetheless, mathematics relies on cognitive mechanisms that were empirically acquired during phylogenetic evolution, and this explains their ‘unreasonable effectiveness’.
Gödel argued that intuition has an important role to play in mathematical epistemology, and despi... more Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light.
The engineering of software is a complex and error prone activity. Software vali- dation and veri... more The engineering of software is a complex and error prone activity. Software vali- dation and verification strives to reduce/eliminate errors in software, thus improving the quality of software. Testing is the most widely used form of software validation and verification. However testing is usually, if not always, a very labor intensive ac- tivity. Despite the fact that most software manufactures
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the fi... more The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.
Logic programming has been introduced as programming in the Horn clause subset of first order log... more Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a non-monotonic reasoning formalism strongly related to Default Logic and Auto-epistemic Logic.
It is proposed that the decision making process is intrinsically formulative in nature; i.e., for... more It is proposed that the decision making process is intrinsically formulative in nature; i.e., for the individual the crux of the process is in the development of a frame, a knowledge base, that can be used to make the decision, rather than the evaluation of the alternatives ’per se’. A two stage model of decision making is forwarded. In the first stage the individual develops his frame. This is the critical, formulative, stage and dependent on the socio‐cultural environment. In the second stage the individual makes a decision by evaluating his frame. This stage is mechanical, evaluative, and determined by the frame formulated during the first stage. A consequence of the basic thesis is that the social and cognitive processes can not be decoupled if we are to understand decision making behavior. However, there are few methods currently available that allow the researcher to look at the relationship between the social and the cognitive process.Herein, a set of methods that permit the researcher to look at this relationship are described and then applied to a particular case study. On the social side these methods are based on a network representation of social structure. On the cognitive side these methods are based on a set of semantic models of human knowledge. A set of cognitive models for representing information are presented — these include frames, maps, and knowledge bases. A set of programs, Frame Technology, for coding, manipulating, and analyzing frames, based on these models, are briefly described. The relationship between social structure and individual cognitive formulations for a group of undergraduates at MIT formulating the problem of selecting a new tutor is then examined using these programs.This work is an overview of a new approach to studying decision making, one that tries to relate social structure to cognitive structure at an empirical level. It is not a detailed analysis of a particular decision: the point is not what decision was made, but the role that information played in this decision. Nor is a detailed discussion of the set of new methods provided: the point is not how the methods work, but the way in which they can be used.
Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of hist... more Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have ... more In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be extended to tensors. The concept of rank is itself difficult to define, and its calculation raises difficulties. Numerical algorithms have nevertheless been developed, and some are reported here, but their limitations are emphasized. These reports hopefully open research perspectives for enterprising readers.
Evolutionary psychology can shed light on mathematical abilities. To do so, it must be supplied w... more Evolutionary psychology can shed light on mathematical abilities. To do so, it must be supplied with a strong model of cultural evolution. Natural selection endowed us with some modules, like the number sense or the logical module, useful for mathematics. Then cultural evolution tinkered with them to create modern mathematics. Of particular significance during cultural evolution is ancient Greece, where a feedback loop that stressed the importance of formal proofs was engaged. Ever since, mathematicians have been engaged in a quest for relevance that uses formal proofs as an indicator of relevance. This led to modern mathematics. Nonetheless, mathematics relies on cognitive mechanisms that were empirically acquired during phylogenetic evolution, and this explains their ‘unreasonable effectiveness’.
Gödel argued that intuition has an important role to play in mathematical epistemology, and despi... more Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light.
The engineering of software is a complex and error prone activity. Software vali- dation and veri... more The engineering of software is a complex and error prone activity. Software vali- dation and verification strives to reduce/eliminate errors in software, thus improving the quality of software. Testing is the most widely used form of software validation and verification. However testing is usually, if not always, a very labor intensive ac- tivity. Despite the fact that most software manufactures
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the fi... more The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.
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