Why you can't prove a single mathematical thing beyond a shadow of a doubt

In summary, the conversation discusses the topic of mathematics and how it is not logically possible to prove its system. The speaker argues that humans live in a reality defined by their perceptions and that this reality is constantly changing. They also mention the idea of a maleficent being manipulating the truth of mathematics, making it impossible for humans to say it is 100% certain. The speaker invites thoughts and acknowledges that it does not meet the minimum posting requirements for the philosophy forum.
  • #1
leemarvin
2
0
Oyez!

I'm an amateur philosopher and I'd like to start up an interesting debate. The topic will be mathematics (and, in general, everything) and how it is not logically possible to prove such a system. This is my informal argument:

1. Humans live in a "reality" governed by laws, axioms, and such.

2. This reality is defined as no more than our perceptions, for our understanding of it depends on our mental processes.

3. This reality is always changing as new studies, etc. change or break certain things.

4. All that is required to make something not 100% certain is a conceivable counter-example.

5. Since humans are not in a position to judge the truth of the world absolutely and without doubt, they must rely on themselves.

6. To borrow Descartes' thought-experiment, if a maleficent being is controlling the world and making sure things abide mathematically only when people are looking, or through some manipulation manages to make the "untruth" that 2+2=4 the truth, how are we to say that mathematics is 100% certain if this random counter-example is even conceivable?

I'd like to hear some thoughts on this. Our own logic dictates that it is not possible to say our logic is 100% sound.
 
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  • #2
This does not meet the minimum posting requirements for the philosophy forum. Please read the guidelines next time.
 

1. Why is it impossible to prove a single mathematical thing beyond a shadow of a doubt?

There are a few reasons why it is impossible to prove a mathematical statement beyond a shadow of a doubt. One reason is that mathematical proofs are based on axioms and rules of logic, which are assumed to be true but cannot be proven themselves. Additionally, the complexity of mathematics means that even the smallest error in a proof can lead to incorrect conclusions. Finally, the concept of infinity in mathematics means that there will always be more unproven statements than proven ones.

2. Can't advanced technology and computers help us prove mathematical statements beyond a shadow of a doubt?

While advanced technology and computers have certainly helped make mathematical proofs more efficient and rigorous, they cannot eliminate the inherent limitations of proving things in mathematics. As mentioned earlier, even the smallest error in a proof can lead to incorrect conclusions, and technology cannot eliminate human error. Additionally, the concept of infinity means that there will always be more unproven statements than can be proven, even with the help of technology.

3. Are there any exceptions to the inability to prove mathematical statements beyond a shadow of a doubt?

There are some mathematical statements that have been proven with absolute certainty, such as the Pythagorean theorem or the irrationality of pi. However, these proofs still rely on axioms and rules of logic that cannot be proven themselves. So while these statements may seem to be proven beyond a shadow of a doubt, there is always a possibility that new information or discoveries could challenge these proofs in the future.

4. How do mathematicians ensure the validity of their proofs?

Mathematicians use rigorous methods and logical reasoning to ensure the validity of their proofs. They also rely on peer review and collaboration with other mathematicians to check for any errors or oversights. In some cases, mathematicians may also use computer programs to help identify potential errors in their proofs.

5. Is the inability to prove mathematical statements beyond a shadow of a doubt a weakness of mathematics?

No, the inability to prove mathematical statements beyond a shadow of a doubt is not a weakness of mathematics. It is simply a reflection of the complex and infinite nature of mathematics. The reliance on axioms and rules of logic is what gives mathematics its strength and allows for the development of new theories and discoveries.

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