Logic and the universe (non-theological question)

[Qoeleth]

It seems that ‘logic’ and mathematics can be understood as linguistic principles. The most basic rule of logic, the principle of non-contradiction, is linguistic (Contra-diction, i.e. speaking in contrary fashion). Also, the truth of 1+1=2 seems to be entirely a linguistic matter, depending upon the definitions of each of the digits and symbols in the formula. On a side note, is it conceivable that there be alternative grammars of logic, in which, for example contradiction may be permitted?

Now, if logic is set of linguistic principle (a principle of the logos/word) why should it follow that it apply in re, in the physical universe? Is there a necessary relationship between the ‘language game’ of mathematics and reality? The existence of ‘imaginary numbers’ suggest that the relationship may not really be one of direct description.

In other words, why should the universe behave logically or mathematically?

In ''real life", we offer encounter questions where the best answer is “Yes and no”. This seems to show the universe may not, in fact, be logical.

 

[itullian]

it seems like chaotic organization to me.

 

[JackHandy]

Hi, just wanted to give a little info regarding something in your post. To clarify, there really is no such thing as an “imaginary” number. What you are referring to was called imaginary by mathematicians who thought that working with square roots of negative numbers was something of lunacy. There is in fact a one to one correspondence between the mathematics of complex numbers and the physical world. Here are a couple of examples that you might find interesting. Do an internet search on each.

(1) Phasors (not Star Trek) used in Electrical engineering

(2) Schrodinger wave equation and its solutions (which are waves).

In both cases, part which is called "imaginary " is no more or less than a phase shift.

This is a long winded (and hopefully interesting) way of saying that yes there is in fact a direct correspondence between physical laws and mathematics.

Check it out and God Bless

 

[Qoeleth]

Hi, just wanted to give a little info regarding something in your post. To clarify, there really is no such thing as an “imaginary” number. What you are referring to was called imaginary by mathematicians who thought that working with square roots of negative numbers was something of lunacy. There is in fact a one to one correspondence between the mathematics of complex numbers and the physical world. Here are a couple of examples that you might find interesting. Do an internet search on each.

(1) Phasors (not Star Trek) used in Electrical engineering

(2) Schrodinger wave equation and its solutions (which are waves).

In both cases, part which is called "imaginary " is no more or less than a phase shift.

This is a long winded (and hopefully interesting) way of saying that yes there is in fact a direct correspondence between physical laws and mathematics.

Check it out and God Bless

Many thanks. I checked out the Wikipedia articles on those things. I was totally lost, but will take your word for it!

May I ask, did this mathematics evolve to describe observable realities, or through abstract mathematical explorations?

Are mathematical formulas ever derived simply from trial and error, or observation and measurement of physical realities?

 

[Alberti_Devoveo]

Hi, just wanted to give a little info regarding something in your post. To clarify, there really is no such thing as an “imaginary” number. What you are referring to was called imaginary by mathematicians who thought that working with square roots of negative numbers was something of lunacy. There is in fact a one to one correspondence between the mathematics of complex numbers and the physical world. Here are a couple of examples that you might find interesting. Do an internet search on each.

(1) Phasors (not Star Trek) used in Electrical engineering

(2) Schrodinger wave equation and its solutions (which are waves).

In both cases, part which is called "imaginary " is no more or less than a phase shift.

This is a long winded (and hopefully interesting) way of saying that yes there is in fact a direct correspondence between physical laws and mathematics.

Check it out and God Bless

The Schrödinger equation requires the complex term in the time-differential component because the wave function is complex. The wave function is complex because quantum mechanics requires single valued functions (meaning, P(x)=P(x+L) where L is the length of the system). Note, however, that when measuring in quantum mechanics, the complex conjugate of the wave function is multiplied to the wave function, resulting in a real value, not imaginary.
Phasors are complex for the exact same reason as the wave function: single values.

In reality, there is nothing “real” about “imaginary” terms. The “imaginary” terms are canceled out from the function when comparing to something in this world. It is not mathematically possible to take sqrt(-1), this term is just represented by i; since it is not mathematically possible to take the square-root, then it must not be physical (real).
Further, the terms “real” and “imaginary” are colloquial terms to represent non-complex and complex terms, respectively. There is absolutely no need to introduce Descartes’ dislike for the impossibility of taking the square-root of a negative number.

 

[Qoeleth]

The Schrödinger equation requires the complex term in the time-differential component because the wave function is complex. The wave function is complex because quantum mechanics requires single valued functions (meaning, P(x)=P(x+L) where L is the length of the system). Note, however, that when measuring in quantum mechanics, the complex conjugate of the wave function is multiplied to the wave function, resulting in a real value, not imaginary.
Phasors are complex for the exact same reason as the wave function: single values.

In reality, there is nothing “real” about “imaginary” terms. The “imaginary” terms are canceled out from the function when comparing to something in this world. It is not mathematically possible to take sqrt(-1), this term is just represented by i; since it is not mathematically possible to take the square-root, then it must not be physical (real).
Further, the terms “real” and “imaginary” are colloquial terms to represent non-complex and complex terms, respectively. There is absolutely no need to introduce Descartes’ dislike for the impossibility of taking the square-root of a negative number.

I am not much of a mathemtician, but it seems even simple negative number don’t really exist. I mean, you can’t have a bag contain -5 apples, can you?

 

[Eleve]

On the other hand, why assume that what it means for a number to exist is for it to count a set of physical objects?

 

[Qoeleth]

On the other hand, why assume that what it means for a number to exist is for it to count a set of physical objects?

Well, the ‘number’ exists but is not instantiated in physical reality?

Is this the same mode of existence as a unicorn?

 

[Eleve]

No, I’d say it’s a similar mode of existence to ethical norms.

Consider this: it’s common in math to say things like “There exists an infinite set that cannot be put into one-to-one correspondence with the natural numbers.” I don’t think anyone would say that such a set exists physically. But it seems wrong to say that those mathematicians are mistaken.

(Additionally, I’d say that having five apples is not the same as instantiating the number five. I don’t have the number five in a bag; I have five apples.)

 

[Qoeleth]

No, I’d say it’s a similar mode of existence to ethical norms.

Consider this: it’s common in math to say things like “There exists an infinite set that cannot be put into one-to-one correspondence with the natural numbers.” I don’t think anyone would say that such a set exists physically. But it seems wrong to say that those mathematicians are mistaken.

(Additionally, I’d say that having five apples is not the same as instantiating the number five. I don’t have the number five in a bag; I have five apples.)

So, if there are five apples in one bag, and five oranges in another- what is there in common? Is the number five simply a ‘name’, a mental concept?

Is mathematics simply a mind game? Are number adjectives or nouns? If I cannot have ‘5’ without having 5 ‘somethings’ it suggests they are adjectives.

In ‘pure’ mathematics, are we then dealing with numbers as substantive adjectives?

 

[Alberti_Devoveo]

I am not much of a mathemtician, but it seems even simple negative number don’t really exist. I mean, you can’t have a bag contain -5 apples, can you?

True, one cannot possess a negative number of objects. In a similar manner, one cannot possess either non-integer or irrational number of objects. You cannot have pi apples (though you can have apple pie ), nor can you have one-half a book.