Why mathematics is neither absolutely nor objectively "right."

[Unknown Soldier]

2 + 2 = 4 means a linear addition of objects.

Actually, 2 + 2 = 4 modulo 5 too. Try this modular arithmetic calculator. Just enter 2 + 2 and modulus 5.

Modulo arithmetic does not represent linear addition.

4 mod 3 + 4 mod 2 = 1

That's about right. What 2 + 2 equals depends on the rules being employed by the mathematician. It's what I've argued from the OP.

I had limited use for mod arithmetic when doing analysis. Differential equations, complex variables,.

Curl, the divergence theorem were very important and of practical utility. Control systems.

That's a good resume, but please understand my main point that I highlighted in bold.

You mean the objective and absolute rules?

Oh sure--just take your pick of which "absolute and objective" rules you like. Other people can take their pick too.

It sure looks to me that many people are seeking absoluteness and objectivity in mathematics when they fail to find it in religion.

 

[steve_bank]

Oh pshaw.

You have an implicit trust in the 'absolute' unambiguous nature of math every time you hit the brakes on your car or fly on a jet or tirn on a light switch or use yourr computer .

I think it was Hilbert in the early 20th century at a conference asked the hypothetical are all mathematical truths povable. There was an exercise to go through math looking for any inconsistencies down to the foundations.

I think it was repeated later in the century.

Yiu probably have no idea what any ambiguities would mean right down to daily tech based life.

 

[Jimmy Higgins]

2 + 2 = 4 means a linear addition of objects.

Actually, 2 + 2 = 4 modulo 5 too. Try this modular arithmetic calculator. Just enter 2 + 2 and modulus 5.

Modulo arithmetic does not represent linear addition.

4 mod 3 + 4 mod 2 = 1

That's about right. What 2 + 2 equals depends on the rules being employed by the mathematician. It's what I've argued from the OP.

I had limited use for mod arithmetic when doing analysis. Differential equations, complex variables,.

Curl, the divergence theorem were very important and of practical utility. Control systems.

That's a good resume, but please understand my main point that I highlighted in bold.

You mean the objective and absolute rules?

Oh sure--just take your pick of which "absolute and objective" rules you like. Other people can take their pick too.

If people solve the same problem using the particular rules for that method and get the same results, that'd imply the rules are objective and absolute. The rules of a sport are often subjective. The rules in math, not so much. Unless one wants to take the meaning of absolute and objective and violate them terribly.

Or are you one of those people that create those stupid Facebook posts asking what 32 plus 17 divided by half equal, and think you've broken math?

 

[Unknown Soldier]

If people solve the same problem using the particular rules for that method and get the same results, that'd imply the rules are objective and absolute.

I'd say that that suggests that people are starting with the same rules and using inference to arrive at the same conclusions. The rules are merely common rather than objective and absolute.

Take the Pythagorean Theorem, for instance. In addition to Hellenistic Greece, this theorem appears to have been arrived at in other places like India and China. Did people all over the world find the Pythagorean Theorem in some mysterious realm of absolute truth? I think it's more likely that right triangles, being handy geometric shapes, were invented and upon examination were understood to have the square of the measure of the side opposite the right angle equal to the sum of the squares of the measures of the other two sides.

The rules of a sport are often subjective. The rules in math, not so much. Unless one wants to take the meaning of absolute and objective and violate them terribly.

Actually, the rules in math can be altered at the whim of the mathematician. Consider imaginary numbers, for example. The rules governing the set of real numbers do not allow for √(-1). However, a need for such a number arose, and it was invented and defined as i = √(-1). No sweat. Nature did not forbid it.

Or are you one of those people that create those stupid Facebook posts asking what 32 plus 17 divided by half equal, and think you've broken math?

No. I'm a mathematician and have studied these issues for decades. I've found it very common for people to think there's something magical about mathematics. But actually mathematics was invented to get work done--work that is peculiar to people. Math is merely a language and methodology we use to describe and work with all those shapes, measures, quantities, and orders that we fuss over.

 

[steve_bank]

Both on and off the form I have heard 'math and science don't get it, 'people don't get it'.

I am a pragmatist, and yes some people think math is magic. Some people think they are magical getting things mere mortals do not.

If yiu are a mathemetcian and loonking at the fundations of math and do not know Peano, counting, arithmetic, and integers then you don't know much.

It is like somenody claiming to be a phsyicst not kbwing Newton's Laws and the Laws Of Thermodynamics.

You said your eduction was psycholgy, and you have posted an edles stearm of philosophizing on yiur threds over the same thing.

Your posting on math is subjective philosophical, not mathematical.


Geodel in his Incompleteness Theorem said in any local consistent system there will always be truths unprovable in the system.

Euclidean Geometry is based on an infinitly small massless point, a line comrised of an infinite numer of points, and the shortest distance between two points is a straight line. None of which are provable, they are defintions. Apllying the rules and defintions of geomerty and the aswer wil always be the same regardless of how itis applied. Logcally constent.

In te end the proof of math is as empirical as is pysical science.We kbow calculus and differntial euaions work n because itis bn out by actual usage.

2 + 2 = 4.

Prove that for any integer n, n + 1 is greater than n ......

Prove that a*(x + y) = a*x + a*y

Prove that (x + y)^2 = x^2 + 2xy + y^2

Prove that (x + y)^-2 = 1/(x + y )^2

Proof such that the statements are absolutely true with no posibility of exceptions.

 

[Cheerful Charlie]

Euclid's points and lines were good enough for geometers of his day. With analytical geometry of Fermat and Descartes, those assumed ideas were sharpened. Points are not things. They are locations on a 2 dimensional Cartesian plane. A line is a set of locations between two points, locations. Described by Cantor's set theory. A triangle's vertex is a location on a plane. Extending Cartesian analytical geometry to 3 dimensions is easy. And below all of that is mathematics. On a number line of an X or Y axis, 2 + 2 = 4. And now on to non-Euclidian geometry and physics. Hilbert spaces, manifolds and more.

 

[Swammerdami]

I cannot endorse the discussions in this thread, but they are vaguely related to the conflicting schools of mathematical philosophy. I've appended a few insightful quotations from pre-eminent mathematicians.

Formalism

Mathematics is a game played according to certain simple rules with meaningless marks on paper.

The essence of mathematics lies in its freedom.

Simplicity

My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful.

Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

Mathematics is the art of giving the same name to different things.

Scientific Mathematics

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength.

Mathematics is the language in which God has written the universe.

To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.

Intuition and Truth

But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.

The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it.

All we know of the truth is that the absolute truth, such as it is, is beyond our reach.

We arrive at truth, not by reason only, but also by the heart.

~ ~ ~ ~ ~ ~ ~
In other news, today or tomorrow I will start a thread with a title like "Did Jesus of Nazareth exist? How to estimate the probability." I hope our mathematicians will read the thread, asking intelligent questions and making other contributions.

~ ~ ~ ~ ~ ~ ~

Geodel in his Incompleteness Theorem said in any local consistent system there will always be truths unprovable in the system.

Nitpick: Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic.

 

[steve_bank]

Euclid's points and lines were good enough for geometers of his day. With analytical geometry of Fermat and Descartes, those assumed ideas were sharpened. Points are not things. They are locations on a 2 dimensional Cartesian plane. A line is a set of locations between two points, locations. Described by Cantor's set theory. A triangle's vertex is a location on a plane. Extending Cartesian analytical geometry to 3 dimensions is easy. And below all of that is mathematics. On a number line of an X or Y axis, 2 + 2 = 4. And now on to non-Euclidian geometry and physics. Hilbert spaces, manifolds and more.

My piunt to Marvin was that a proof exists in context of a set of rules.

 

[steve_bank]

I cannot endorse the discussions in this thread, but they are vaguely related to the conflicting schools of mathematical philosophy. I've appended a few insightful quotations from pre-eminent mathematicians.

Formalism

Mathematics is a game played according to certain simple rules with meaningless marks on paper.

The essence of mathematics lies in its freedom.

Simplicity

My work always tried to unite the Truth with the Beautiful, but when I had to choose one or the other, I usually chose the Beautiful.

Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

Mathematics is the art of giving the same name to different things.

Scientific Mathematics

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength.

Mathematics is the language in which God has written the universe.

To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls.

Intuition and Truth

But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.

The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there never was any other object for it.

All we know of the truth is that the absolute truth, such as it is, is beyond our reach.

We arrive at truth, not by reason only, but also by the heart.

~ ~ ~ ~ ~ ~ ~
In other news, today or tomorrow I will start a thread with a title like "Did Jesus of Nazareth exist? How to estimate the probability." I hope our mathematicians will read the thread, asking intelligent questions and making other contributions.

~ ~ ~ ~ ~ ~ ~

Geodel in his Incompleteness Theorem said in any local consistent system there will always be truths unprovable in the system.

Nitpick: Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic.

Yes. Math and Euclidean geometry are consistent, if not it would be useless.

There is a sayng I heard from time to time when technical debate got personal.

'Science always works'. Meaning that regardless of what you think a problem comes down to science and mathematical theories.

 

[Swammerdami]

Geodel in his Incompleteness Theorem said in any local consistent system there will always be truths unprovable in the system.

Nitpick: Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic.

Yes. Math and Euclidean geometry are consistent, if not it would be useless.

In nitpicking your claim, I may have misplaced the emphasis and made your omission unclear. Let me try again:

Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic. (And as far as I know, the consistency of such systems has NOT been proven.)

 

[Unknown Soldier]

Euclid's points and lines were good enough for geometers of his day.

What do you mean by "good enough"?

With analytical geometry of Fermat and Descartes, those assumed ideas were sharpened.

Actually, Euclid's geometry was never "sharpened" by René Descartes, whatever that might mean. Euclid's geometry is as useful as it ever was. However, what René Descartes did do was to create a planar coordinate system that defined points as ordered pairs on a plane centered on the point (0, 0), the "origin," and divided into four quadrants divided by a horizontal axis and a vertical axis both of which are intersected by the origin. The advantage of this Cartesian coordinate system is that it allows for geometric shapes created by relations between the coordinates. That way geometric shapes can be better analyzed using algebraic expressions like y = x/2.

Points are not things.

Points are things.

They are locations on a 2 dimensional Cartesian plane.

Points can be locations on the Cartesian plane, but not necessarily. Points can be plotted in three dimensions, for example, or points can be defined in any higher number of dimensions. Simply put, points are positions.

A line is a set of locations between two points, locations.

Technically, a line is of infinite length and so cannot be constructed between two points. Rather, a line segment's length is the shortest distance between two points, the line segment's "end points." A lot of people incorrectly say a line is the shortest distance between two points, but a line is not a distance but an infinite set of points.

A triangle's vertex is a location on a plane.

Not necessarily. Triangles can be constructed on the surface of a sphere.

Extending Cartesian analytical geometry to 3 dimensions is easy. And below all of that is mathematics.

Below all of what is mathematics?

On a number line of an X or Y axis, 2 + 2 = 4.

On the real number line 2 + 2 = 4, but number lines can be made circular in which 2 + 2 ≠ 4.

And now on to non-Euclidian geometry and physics. Hilbert spaces, manifolds and more.

With all due respect, you're just rambling and posting a lot of errors as a result. I'd recommend that if you want to know something about math, then study it more.

 

[Cheerful Charlie]

[removed] 2 + 2 still equals 4.

 

[steve_bank]

Geodel in his Incompleteness Theorem said in any local consistent system there will always be truths unprovable in the system.

Nitpick: Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic.

Yes. Math and Euclidean geometry are consistent, if not it would be useless.

In nitpicking your claim, I may have misplaced the emphasis and made your omission unclear. Let me try again:

Gödel's Incompleteness Theorem guarantees unprovable truths only in consistent systems capable of modeling basic arithmetic. (And as far as I know, the consistency of such systems has NOT been proven.)

I do not dispute your right to nitpick , my friend. I have probably nitpicked you in the past.

It depends on what you mean by proven. That is why I think in the end math is as empirical as physical science.

Wiithin bounds, not quantun and nor relativistc, Newton's Laws have been demonstrated to the extent no one questions the validity.

Within bounds Euclidean geometry has been used to the extent it not questioned.

There is always the possibility that somewhere in all of math there is some inconsistency.

I look at Euclidean geometry lie a syllogism. Given the premises of a point, line, and shortest distance the conclusion of rules of geometry follow.

To me math works because it is designed to work, no different tan designing a machine. Math is not discovered it is designed. To me a tool in principle the same as a screwdriver.

I belive all math comes down to counting, subtraction, and addition. Integral calculus is an arithmetic summation. Differntial ca;cuus is a subtracton taken to a limit.

I'd have to look it up, I believe addition is defined as a field. Multiplication and division are addition and subtraction.

In Systems International it is al based on the definition of meters, kilograms, and seconds. Arbitrary defintions.

 

[steve_bank]

The real number line is a line. A circle is not a line.

The top level set of numbers is C complex numbers.

z = x +- iy. Real plus Imaginary numbers where i is sqrt(-1). Where x is on the real axis and y the imaginary axis of an xy linear coodinate system.

A real number is a complex number with a zero imaginary part. z = x + i0;

MaGNITUDE z = sqrt(x^ + y^2) a vector wit the origin at (0,0). Trigonometry.

 

[Unknown Soldier]

[removed]

Whew! It's nice to argue my case free of insults.

2 + 2 still equals 4.

Check the attached image of David Poole's Linear Algebra pages 14 - 15 in which he explains how the sum 2 + 2 can vary.

 

[Unknown Soldier]

The real number line is a line. A circle is not a line.

Check the attached image of David Poole's Linear Algebra pages 14 - 15 in which he not only explains how the sum 2 + 2 can vary but how a number "line" can be circular..

 

[Cheerful Charlie]

Linear algebra does not magically change 2 + 2 = 4 to 2 + 2 =/= 4.

 

[steve_bank]

Quote mining. Again misreading based on your lack of math fundamentals.

Ah yes linear algebra AX=B. Used it to solve simultaneous equations.

The definition of a circle is a set of points equidistant from a point. Or the locus of points equidistant from a point.

I would say it is more correct to say if a circle is continuous it has an infinite number of points. A circle can be integer points. To me saying a line can be made into a circle is a manner of speaking. Unless possibly we are talking about topological transformations, not linear algebra.

https://www.berkeleycitycollege.edu/wjeh/files/2013/01/geometry_note_circle.pdf

Definition: A circle is the set of all points in a plane that are equidistant from
a given point called the center of the circle. We use the symbol to represent
a circle

 

[steve_bank]

Marvin,

I know the general undergrad applied math and maybe a little more and have used a lot of it, but I'd be embarrassed to call myself a mathematician.

So, you read some books and wtaced videos and you are a mathematician?

Going back aways I watched MIT's Starng's vdeo recorded Linaear Algebra clases and got his book,

I recommend it. If you eant a math self education it is all online at MIT.

Linear Algebra | Mathematics | MIT OpenCourseWare

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

ocw.mit.edu

 

[Unknown Soldier]

Marvin,

Marvin?

I know the general undergrad applied math and maybe a little more and have used a lot of it, but I'd be embarrassed to call myself a mathematician.

I understand why.

So, you read some books and wtaced videos and you are a mathematician?

Yes, I've studied plenty of books, and I recommend reading them. My intensely studying many math books both for my formal college education and through self study has given me the math knowledge I have today.

Speaking of reading, did you read the attached file? It clearly states that given the integers modulo 3,
2 + 2 = 1, not 4 like you keep asserting. Also, regarding the number line for arithmetic modulo 3, Poole says: "We can visualize the (integers modulo 3) number line as wrapping around a circle..." So contrary to what you assert, number "lines" are not necessarily the straight lines from geometry.

So I'm right on both counts and you're wrong on both counts. The difference is clearly based on my having a respect for the discipline of mathematics and for math education while you are espousing a strange sort of fundamentalism in which you must believe what you want to believe about math. You can believe whatever you wish about anything, but what you've been posting in this forum does a disservice to math education. There's enough math illiteracy as it is, and it won't do the situation any good by posting a pet philosophy that masquerades as a useful understanding of mathematics.