Do Mathematical Truths Depend on the Existence of Mathematical Objects?

[poverlord]

Hey guys, I have been thinking about the problem of mathematical truths. We simply do not know if the things that mathematics deals with really exist or not, yet people still believe that its statements are true and indubitable. Unfortunately, the only truly well-respected theory of truth in the sciences requires some notion of sentences "corresponding" to the real world. If it turns out that mathematical "objects" don't exist then what can we say about mathematical truths. In short, how can we say true things about stuff that does not exist?

 

[waht]

The concept of a 'mathematical truth' ,like any other concept, is generated by billions of interconnected neurons in the brain that are physical, and follow laws of physics, and chemistry. So in a sense, a 'mathematical truth' is encoded in a physical medium which allowed it to be so.

 

[Greg Bernhardt]

Please read this thread and repost accordingly
https://www.physicsforums.com/showthread.php?t=459350

 

[HallsofIvy]

Hey guys, I have been thinking about the problem of mathematical truths. We simply do not know if the things that mathematics deals with really exist or not, yet people still believe that its statements are true and indubitable. Unfortunately, the only truly well-respected theory of truth in the sciences requires some notion of sentences "corresponding" to the real world. If it turns out that mathematical "objects" don't exist then what can we say about mathematical truths. In short, how can we say true things about stuff that does not exist?

All mathematical statements are of the form "If ... then ..." (even if the "if" part is not explicitely stated). That is, mathematics is about "structure", not "content". Think of mathematics as "templates" where you enter the specific content into the "blanks".

It is true, from the meanings of the words "or" and "if ... then", that "if a is true, then "a or b is true", is a true statement. The "content"- what "a" and "b" mean and whether a is true or not, is not part of that statement and the truth of the statement does not depend on the truth of a and b separately.

 

[LudusRex]

I understand the importance of evidence and empirical data in determining the truth of a statement. However, when it comes to mathematical truths and ontology, the answer may not be as straightforward. While it is true that we cannot physically observe or measure mathematical objects, their existence and truthfulness can still be supported through logical reasoning and consistency with other mathematical principles.

One way to approach this problem is to view mathematical objects as abstract concepts that exist in the realm of thought rather than in the physical world. Just because something does not have a physical form does not mean it does not exist in some form. For example, the concept of infinity may not have a tangible existence, but it is still a fundamental concept in mathematics that allows us to make accurate predictions and understand the world around us.

Additionally, the truth of mathematical statements can also be supported by their usefulness in solving real-world problems and making predictions. For example, the mathematical concept of probability may not have a physical existence, but it has proven to be a valuable tool in predicting outcomes in various fields such as economics, medicine, and engineering.

In short, while the existence of mathematical objects may be debatable, their truthfulness can still be supported through logical reasoning and their practical applications. As scientists, it is important for us to remain open-minded and continue to explore and question the nature of truth and ontology in all fields, including mathematics.