Abstract
A new cognitive foundational program for the fulfillment of artificial mathematical intelligence is proposed: The New Cognitive Foundations for Mathematics. This program aims to ground mathematics and (the corresponding parts of) logic (founding parts of mathematics) from a multidisciplinary perspective. The main challenges and cornerstones of this new program are described. Additionally, fundamental aspects of the cognitive substratum of (current formalizations of the notion of) mathematical proof are analyzed in detail. Finally, basic principles of the local nature of the (conscious) mind are presented where mathematics is considered, to some extent, as an explicit (cognitive) product of it.
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Notes
- 1.
Here it is important to clarify that, implicitly, cognitive science includes, in its original and modern approaches, results and methods from psychology, (neuro)biology, computer science, anthropology, and AI, among others. However, it is always valuable to explore new connections that cognitive science, as a well-established discipline, can have with the latest results of the former disciplines for understanding (the fundamental principles of) the mind, and therefore of mathematical research. Therefore, all the other scientific disciples are explicitly mentioned, although they are partially included in cognitive science.
- 2.
This specific approach can be included as a part of the classic cognitive foundations for mathematics, introduced at the same time in [14].
- 3.
The interested reader can take a look into Chap. 5 concerning the physical numbers, which can be seen as a formal refinement of the natural numbers with a stronger cognitive and physical basis.
- 4.
At this point, it is worth noting that the incompleteness theorems would need, as one of its minimal hypothesis, the existence of infinity structures (e.g., an infinity tree), which are again essentially non-physical statements [4].
- 5.
- 6.
Standard mathematics in this context means all the standard theories that we can ground based on ZFC set theory, which would represent, let us say, more than 95% of modern mathematics.
- 7.
- 8.
Here we mean the formal as well as the informal notation that we use for getting intuitions and for subsequently formalizing (mathematical) arguments.
- 9.
For more about this, see Chap. 12, where we describe the most fundamental future challenges of the AMI (meta-)program.
- 10.
Again, the reader can find more information in Chap. 12.
- 11.
For a more detailed collection of such researchers, together with general considerations about the impact of their work, see [35].
- 12.
This statement has a more explicit content suitable to be combined with additional facts involved in the corresponding conjecture.
- 13.
This is, in fact, one of the common objections that constructive mathematics does to the “standard” proof methods [27].
- 14.
Based on the state of the art of neurobiology and psychology, it is clear that our unconscious mind also plays a primary role in conceptual creation. However, we want to emphasize here the primary features of the conscious part of the mind, which are fundamental during abstract creation as well.
- 15.
The usage of the term “unity” is inspired by the unified nature of conscious experience, as well as for the quest of finding a formal quantification for it [7].
- 16.
This machinery will be developed in much more detail in the second part of the book.
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Gómez Ramírez, D.A.J. (2020). General Considerations for the New Cognitive Foundations’ Program. In: Artificial Mathematical Intelligence. Springer, Cham. https://doi.org/10.1007/978-3-030-50273-7_3
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