Is objective truth mathematics?

[STT]

Which definition of mathematics are you using? There are several different definitions of mathematics, many of which are controversial and not accepted by the majority of mathematicians. Are you a mathematician? If not, then why is your definition of mathematics the one we should accept?
en.wikipedia.org/wiki/Definitions_of_mathematics

Lets define mathematics as a set of mind independent rules between objects with the aim to construct knowledge.

 

[Tomdstone]

Lets define mathematics as a set of mind independent rules between objects with the aim to construct knowledge.

Aristotle does not aggree with that definition. He says mathematics is the science of quantity.

 

[STT]

Aristotle does not aggree with that definition. He says mathematics is the science of quantity.

What is wrong with my definition?

 

[PseuTonym]

Originally Posted by John Martin View Post

“Objective truth” is truth of individual objects.

Objective truth is a set of abstract rules/principals. So I disagree with you.

Can you explain how you see any subjectivity creeping into things merely as a byproduct of focusing on a specific, verifiable counter-example to a conjecture? An alleged proof that a conjecture is true might be defective: at least one step in the reasoning is invalid, or at least one assumption relied upon is actually false.

[In 1988, Roger Frye found the smallest possible counterexample to Euler’s sum of powers conjecture:

(95800 to the power of 4)+(217519 to the power of 4)+(414560 to the power of 4)=(422481 to the power of 4)
](Euler's sum of powers conjecture - Wikipedia)

There are other examples at the following link:

math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples

 

[PseuTonym]

What is wrong with my definition?

You need to begin with a list of all of the basic, undefined terms that are available to use.

Also, you should specify whether you are:

1. stipulating your own new meaning for an existing phrase, or
2. trying to describe an already established meaning of a phrase based on your observation of usage of that phrase by others, and your attempt to understand the meaning from the context.

 

[PseuTonym]

Can you explain how you see any subjectivity creeping into things merely as a byproduct of focusing on a specific, verifiable counter-example to a conjecture? An alleged proof that a conjecture is true might be defective: at least one step in the reasoning is invalid, or not all of the “objective truths” that the proof relies upon are actually true.

 

[Mintaka]

Geometry is a form of logic that deals with ideal forms that don’t exist in the real world, but which can be applied approximately to the non-ideal forms of the real world.

For example, in the real world, there is no such thing as a “point” or a “line,” as mathematics defines them.

 

[PseuTonym]

STT, if you like reading people who agree with you, then you might find the following two essays interesting:

1. “The a priori”, by Alfred Jules Ayer;
2. “On the nature of mathematical truth”, by Carl G. Hempel;

The numbers 17 and 19 are chapter numbers in the following book:

Philosophy of Mathematics: Selected Readings
Paul Benacerraf (Editor), Hilary Putnam (Editor)

Hempel is also responsible for …

en.wikipedia.org/wiki/Raven_paradox

I find it a bit surprising that there is any controversy about how to resolve that so-called paradox.

 

[PseuTonym]

Now lets look at reality and see how well laws of physics describes the state and evolution of Cosmos.

Physics, astronomy, and cosmology are the subjects that deal with the state and evolution of the Cosmos. However, those are very specialized subjects, and a lot of information is simply unavailable. For example, so-called “exo-planets” have been detected near stars other than the Sun, but very little is known about those planets.

If you want to look at reality, then we should be able to focus attention on the Earth, if the Earth is part of reality. Does the Cosmos include the Earth?

Here is a fact:
On July 16, 1945, the Allied Manhattan Project successfully detonated an atomic bomb in the New Mexico desert.

Link:
en.wikipedia.org/wiki/Atomic_bombings_of_Hiroshima_and_Nagasaki

I suppose that you could use physics to describe the detonation of an atomic bomb, but if the same kind of bomb is used, then it will be the same physics at any time and any location. I see no evidence that anybody can use physics to predict when and where an atomic bomb will detonate, the way that astronomy is used to predict when a solar eclipse will occur.

Link:
hsm.stackexchange.com/questions/368/when-did-it-become-possible-to-predict-the-time-and-place-of-solar-eclipses

So what does physics tell us about the evolution of our part of the Cosmos? Physics is like a book of recipes. However, there’s no food in a book of recipes.

An ordinary book of recipes doesn’t even contain maps or addresses to identify where retail stores that sell food are located, or will be located in future as circumstances evolve.

 

[STT]

You need to begin with a list of all of the basic, undefined terms that are available to use.

Also, you should specify whether you are:

1. stipulating your own new meaning for an existing phrase, or
2. trying to describe an already established meaning of a phrase based on your observation of usage of that phrase by others, and your attempt to understand the meaning from the context.

I don’t have any degree on philosophy of math. This is my definition: Mathematics is a set of mind independent rules between objects with the aim to construct knowledge.

Here I offer the meanings of terms:

Object: An abstract entity.
Rule: An operator which makes a relation between objects.
Knowledge: A state of belief which is true.

 

[STT]

Can you explain how you see any subjectivity creeping into things merely as a byproduct of focusing on a specific, verifiable counter-example to a conjecture?

An alleged proof that a conjecture is true might be defective: at least one step in the reasoning is invalid, or not all of the “objective truths” that the proof relies upon are actually true.

This is a good and hard question. I will think about it.

 

[STT]

STT, if you like reading people who agree with you, then you might find the following two essays interesting:

1. “The a priori”, by Alfred Jules Ayer;
2. “On the nature of mathematical truth”, by Carl G. Hempel;

The numbers 17 and 19 are chapter numbers in the following book:

Philosophy of Mathematics: Selected Readings
Paul Benacerraf (Editor), Hilary Putnam (Editor)

Hempel is also responsible for …

en.wikipedia.org/wiki/Raven_paradox

I find it a bit surprising that there is any controversy about how to resolve that so-called paradox.

Thank you.

 

[STT]

Physics, astronomy, and cosmology are the subjects that deal with the state and evolution of the Cosmos. However, those are very specialized subjects, and a lot of information is simply unavailable. For example, so-called “exo-planets” have been detected near stars other than the Sun, but very little is known about those planets.

If you want to look at reality, then we should be able to focus attention on the Earth, if the Earth is part of reality. Does the Cosmos include the Earth?

Here is a fact:
On July 16, 1945, the Allied Manhattan Project successfully detonated an atomic bomb in the New Mexico desert.

Link:
en.wikipedia.org/wiki/Atomic_bombings_of_Hiroshima_and_Nagasaki

I suppose that you could use physics to describe the detonation of an atomic bomb, but if the same kind of bomb is used, then it will be the same physics at any time and any location. I see no evidence that anybody can use physics to predict when and where an atomic bomb will detonate, the way that astronomy is used to predict when a solar eclipse will occur.

Link:
hsm.stackexchange.com/questions/368/when-did-it-become-possible-to-predict-the-time-and-place-of-solar-eclipses

So what does physics tell us about the evolution of our part of the Cosmos? Physics is like a book of recipes. However, there’s no food in a book of recipes.

An ordinary book of recipes doesn’t even contain maps or addresses to identify where retail stores that sell food are located, or will be located in future as circumstances evolve.

That is not quite correct. We can make a lots of prediction using physics, weather forecast. Most of things that we build is based on laws of physics. We however cannot predict earthquake.

 

[Tomdstone]

We can make a lots of prediction using physics, weather forecast.

Weather forecasts are known to be way off sometimes. I was in San Francisco once and the weatherman said it was sunny and clear, and it was going to be that way for a while. However, while he was saying that, it was pouring down rain as I was walking on a street in the center of San Francisco and there was no indication that it was going to stop soon.

 

[PseuTonym]

That is not quite correct. We can make a lots of prediction using physics, weather forecast.

I think that on July 16, 1945, it suddenly got very hot, much hotter than usual, somewhere in the New Mexico desert. However, if the bomb had failed to explode, then there would have been ordinary weather. So, it seems that you need a weather prediction model that …

#1 … has access to information about when and where there will be attempts to detonate bombs, and
#2 … is able to evaluate all of the data to determine whether or not the object will actually explode.

However, if #2 is possible, then there was no need to test the bomb. The explosion in the New Mexico desert on July 16, 1945 was part of a secret, military project. Some civilians heard something and some also saw something. The media received false information from the government to provide a cover story to explain what was otherwise inexplicable.

If one fine day in 1954 you had been at the Bikini Atoll, and suddenly an enormous ball of fire detonated all around you, and melted all of the thermometers in the neighborhood, then wouldn’t you say that the weather had gotten uncomfortably hot?

The second series of tests in 1954 was codenamed Operation Castle. The first detonation, Castle Bravo, was a new design utilizing a dry fuel thermonuclear hydrogen bomb. It was detonated at dawn on March 1, 1954. The 15 megaton nuclear explosion far exceeded the expected yield of 4 to 8 megatons (6Mt predicted), and was about 1,000 times more powerful than each of the atomic bombs dropped on Hiroshima and Nagasaki during World War II. The scientists and military authorities were shocked by the size of the explosion and many of the instruments they had put in place to evaluate the effectiveness of the device were destroyed

Link:
en.wikipedia.org/wiki/Nuclear_testing_at_Bikini_Atoll

 

[STT]

Weather forecasts are known to be way off sometimes. I was in San Francisco once and the weatherman said it was sunny and clear, and it was going to be that way for a while. However, while he was saying that, it was pouring down rain as I was walking on a street in the center of San Francisco and there was no indication that it was going to stop soon.

Yes, but there are two main reasons that why prediction is sometimes off, namely (1) we don’t have enough necessary information (2) equations are chaotic. That doesn’t mean that laws of physics are useless.

 

[STT]

I think that on July 16, 1945, it suddenly got very hot, much hotter than usual, somewhere in the New Mexico desert. However, if the bomb had failed to explode, then there would have been ordinary weather. So, it seems that you need a weather prediction model that …

#1 … has access to information about when and where there will be attempts to detonate bombs, and
#2 … is able to evaluate all of the data to determine whether or not the object will actually explode.

However, if #2 is possible, then there was no need to test the bomb. The explosion in the New Mexico desert on July 16, 1945 was part of a secret, military project. Some civilians heard something and some also saw something. The media received false information from the government to provide a cover story to explain what was otherwise inexplicable.

If one fine day in 1954 you had been at the Bikini Atoll, and suddenly an enormous ball of fire detonated all around you, and melted all of the thermometers in the neighborhood, then wouldn’t you say that the weather had gotten uncomfortably hot?

Link:
en.wikipedia.org/wiki/Nuclear_testing_at_Bikini_Atoll

I agree that lack of enough information can make a model fail to predict the correct answer.

 

[PseuTonym]

What is wrong with my definition?

There is an accident ready to happen when you observe the assignment of an official topic name or label (“mathematics”) to some claims, and express confidence that if the topic label is appropriate, then the claims are true.

Perhaps you should create a new thread for this: how mathematical controversies have been resolved. Errors have been discovered in alleged proofs (published in reputable academic math journals), and in some cases the claims that were allegedly proven are now believed to be false. Of course, there is a difference between an error in reasoning, and an error in the premises. It is conceivable that a published proof from the past could be now discredited simply because of a change in beliefs.

However, ordinarily an error in premises takes the form of a conjecture invoked without recognizing that it is merely a conjecture, such as when an existing theorem is extrapolated via analogy. For example, some attempts to prove Fermat’s Last Theorem failed because a superset of the ring of integers was involved in the attempted proof, and it was tacitly assumed that unique factorization (a theorem for ordinary integers) is also a theorem in the particular ring that was used in the proof.

 

[PseuTonym]

Perhaps a more interesting example of a controversy: some proofs in calculus were discredited on the grounds that they appealed to infinitesimals, but allegedly at least one such discredited proof (written in the 1600s or 1700s) is now considered completely correct, exactly as written, based on definitions that were created by Abraham Robinson in the area of study that is now known as “Non-standard Analysis.”

 

[STT]

There is an accident ready to happen when you observe the assignment of an official topic name or label (“mathematics”) to some claims, and express confidence that if the topic label is appropriate, then the claims are true.

So what is wrong with my definition?

Perhaps you should create a new thread for this: how mathematical controversies have been resolved. Errors have been discovered in alleged proofs (published in reputable academic math journals), and in some cases the claims that were allegedly proven are now believed to be false. Of course, there is a difference between an error in reasoning, and an error in the premises. It is conceivable that a published proof from the past could be now discredited simply because of a change in beliefs.

I am not really that good on math and I don’t think that this forum is appropriate for such discussion.