The Nature and Future of Axiomatic Systems

Dear Colleagues,

The ideal form of the description of mathematical knowledge is generally thought to be the axiomatic approach: the laws of logical inference and a few statements define a mathematical topic.

As the ultimate form of compression, this approach realizes mathematics' efficient elegance and unity. However, there are issues. First, it may be challenging to decompress: finding whether a statement is a logical consequence of the axioms takes work. This endeavor is the realm of automated theorem proving. How to find solid proof by computers is a topic of ongoing research. Moreover, as we have learned in the development of mathematical logic, it could be impossible in some instances. These are the philosophical limits of mathematics, and the ramifications still need to be fully understood.

A recent issue is the involvement of AI-based frameworks (such as Large Language Models) in mathematical discovery and description. Can and will these frameworks follow the axiomatic pathway? We also need to consider the perspective of education. Is the axiomatic treatment the most conducive to understanding? Or, as it may feel to the learner, why do we have islands of concepts connected in tangled webwork but not crystallized from a tiny seed? If understanding does not require solid foundations, is the axiomatic approach simply a historical tradition? 

For this Special Issue, we invite both types of contributions: those that are `inside', e.g., theorem-proving, axiomatic treatment of a mathematical topic; and those from 'outside', e.g., critical philosophical work and papers investigating the educational aspects of axiomatic systems. 

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but not limited to) the following:

• Theorem proving;
• Axiomatic treatment of a mathematical topic;
• Axiomatism and artificial intelligence;
• Educational aspects of axiomatic systems. 

We look forward to receiving your contributions.

Dr. Attila Egri-Nagy
Prof. Dr. Miklos Hoffmann
Guest Editors

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• axiomatic systems
• theorem proving
• artificial intelligence
• mathematical logic
• mathematics education