A problem with the Axiomatic Foundations of Mathematics

In summary, the conversation discusses the possibility of inconsistencies within the axiomatic systems of modern mathematics and how this can affect the validity of proofs and the foundation of the subject. It also mentions the work of Vladimir Vovoedsky and Ed Nelson on alternative foundations for mathematics, and the discussion surrounding them. The conversation concludes with the idea that if math were found to be inconsistent, it would be a significant and interesting development, but math itself would still continue and evolve.
  • #36
Skrew said:
After looking at the axiomatic systems of modern mathematics and asking myself what proves they are self consistent I went looking for an explanation and so far I have found only that they have not been proven self consistent nor likely will a proof ever exist. So the possibility of a contradiction being present within the system exists. Therefore when using the system you assume it has no contradictions present within it.

I find this incredibly disturbing as it means every proof I have written would become worthless should the axiomatic system it is written in be demonstrated to be inconsistent.

I find this so disturbing that I question if I want to pursue my studies in mathematics, one thing I always liked about mathematics is that I considered it built on unshakable ground but this appears not to be the case.

Has anyone else experienced this revelation? How do you deal with it?

Man is not an absolute. We build in our own limits and mistakes.
Your ability to see this situation is why you should stick with it.
Others have advised you to look into formal logic etc.
I would add a good lesson or book on history of the subject.
There also exist papers wrote out there covering the same questions and more by noted mathematicians.
Finding of such may also be of value to you and your decisions.
 
<h2> What is the Axiomatic Foundations of Mathematics?</h2><p>The Axiomatic Foundations of Mathematics is a set of fundamental principles or axioms that serve as the basis for all mathematical reasoning and proof. These axioms are considered to be self-evident and do not require any further justification.</p><h2> What is the problem with the Axiomatic Foundations of Mathematics?</h2><p>The main problem with the Axiomatic Foundations of Mathematics is that it is based on a set of axioms that cannot be proven or verified. This means that the entire structure of mathematics is built on assumptions that may or may not be true.</p><h2> How does this problem affect mathematics?</h2><p>This problem can lead to inconsistencies and contradictions within mathematical systems. It also means that there may be different sets of axioms that can be used to build mathematical systems, leading to debates and disagreements among mathematicians.</p><h2> Are there any proposed solutions to this problem?</h2><p>Yes, there are several proposed solutions, such as using different sets of axioms or developing new axioms that can be proven. Some mathematicians also argue for a more flexible approach to axioms, allowing for revisions and updates as needed.</p><h2> How does this problem impact other fields of study?</h2><p>The Axiomatic Foundations of Mathematics is the basis for many other fields of study, such as physics, engineering, and computer science. Therefore, any issues with the foundations of mathematics can have a ripple effect on these fields and potentially impact our understanding of the world around us.</p>

FAQ: A problem with the Axiomatic Foundations of Mathematics

What is the Axiomatic Foundations of Mathematics?

The Axiomatic Foundations of Mathematics is a set of fundamental principles or axioms that serve as the basis for all mathematical reasoning and proof. These axioms are considered to be self-evident and do not require any further justification.

What is the problem with the Axiomatic Foundations of Mathematics?

The main problem with the Axiomatic Foundations of Mathematics is that it is based on a set of axioms that cannot be proven or verified. This means that the entire structure of mathematics is built on assumptions that may or may not be true.

How does this problem affect mathematics?

This problem can lead to inconsistencies and contradictions within mathematical systems. It also means that there may be different sets of axioms that can be used to build mathematical systems, leading to debates and disagreements among mathematicians.

Are there any proposed solutions to this problem?

Yes, there are several proposed solutions, such as using different sets of axioms or developing new axioms that can be proven. Some mathematicians also argue for a more flexible approach to axioms, allowing for revisions and updates as needed.

How does this problem impact other fields of study?

The Axiomatic Foundations of Mathematics is the basis for many other fields of study, such as physics, engineering, and computer science. Therefore, any issues with the foundations of mathematics can have a ripple effect on these fields and potentially impact our understanding of the world around us.

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