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Handling mathematical concepts that have no relation to reality

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brian0918
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Or even more simple, take exponents. 2^2 apples is 2 sets of 2 apples. Well, that's 4 apples. 1^3 apples is one set of (one set of one apple) which is just one apple. But now what about 2^0? Zero sets of two apples? Well, that is one apple. And 2^-1? A negative set of two apples? Well, that is half an apple.
You’re fostering the same problem you’re trying to correct. Explain to a kid why zero sets of one apple is not zero, or how many sets of sets 2^(3/2) equals. I don’t think traditional methods of teaching imaginary and negative numbers have any less relation to reality than yours do.

The important similarity between your method and traditional methods is that they are correct within certain situations – they are contextual, and that’s not exactly a problem that we as philosophers can (or would even want to) solve.

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Hmm, I just need to clarify my point. Your example is true, but that's not what I meant. For instance, a one dollar bill physically exists. It is an entity. And there is one of those bills. But you cannot have -1 of those bills. Do you understand what I mean? The least quantity any entity can be is 1. If an entity has 0 quantity, then it does not exist. Therefore, in this sense, negative quantities are an impossibility and would imply some alternative to existence vs. non-existence, like...less than non-existence.

Even when your liabilities exceed your assets, the fact that your net worth is a negative number is not an actual entity. This is still a concept of method. Physically what is going on is you lose all your assets, but you don't continue losing assets once you go negative. You have no more assets to lose when you have 0 assets! No, what happens is you go in debt. You don't have anymore assets, so you are going to have to get more in the future. I hope my point is clear.

Your point is clear, but it does not support a conclusion that negative numbers are somehow invalid. I suggest you read what has been written in this thread about the validity of negative numbers and understand why they are not only perfectly valid but also absolutely necessary. What you have written above seems to suggest that only the counting numbers (i.e., 1, 2, 3, 4, 5, ...) are valid. Maybe you would also accept rational numbers formed from the counting numbers? If we (humankind) were to accept only counting numbers and rational numbers formed from the counting numbers we would set science and technology backwards many centuries!

John Link

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Here's how I think of exponents.

Addition is adding numbers together. Easy to validate to sense perception.

2 + 4 = 6.

Multiplication is a "step up" from addition. Mutliplying a number is just adding it to itself a bunch of times.

2 * 4 = 2 + 2 + 2 + 2 = 8

I think of exponents as a "step up" from that.

24 = 2 * 2 * 2 * 2 = 16

Are you seriously saying that they didn't teach it to students this way? How old are you? Schools must be even worse now than when I was in school. I had a huge problem with square roots until I finally realized that they were just the opposite of squaring, much like division is the opposite of multiplication. It wasn't explained to me. I had to eventually figure it out on my own.

I hate math. I honestly hate it. I have ever since high school. I only know what I have to know to be a practical programmer.

Yet in elementary school, I was so good at it, me and three or four other students in my school got to start learning basic algebra before everyone else. Once I got to high school, I'm not certain how, but it became different. It was no longer interesting or fun as it became increasingly, nonsensically complex.

If you've been paying attention to the 1 = .9999... repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.

Edited by Amaroq
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Obviously there is no such quantity as -1 in real life, ..

Neither is there a quantity as +1 in real life. Things like 1 cow or 1 dollar do exist in real life, but not the number 1. The latter is an abstraction that can be useful in counting real things, so that an abstract concept like natural number has a useful application in real life. Exactly the same holds for other abstract notions like negative numbers, rational numbers, real numbers or complex numbers. These can all be used for real-life applications, in the case of negative numbers for example for bookkeeping (an extension of the notion of counting) and in the case of complex numbers for example for electrical circuits. In general the connection between the abstract concept of a certain kind of number will be more "complex" (no pun intended) for more general categories, like complex numbers, than for natural numbers (counting things). That in the latter case the link between number and real object is simple (counting) does not mean that the number (an abstract concept) and the object (cow, dollar, real objects) are the same. So when you realize that all numbers are abstract and there doesn't exist any number in reality, but that those numbers and the corresponding theories can be applied to real life situations, there isn't any problem.

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If you've been paying attention to the 1 = .9999... repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.

I haven't followed that thread, but the fact that 1 = .9999...repeating follows by elementary logic from the mathematical definitions and notation, it's no nonsense, on the contrary, it's the only logical answer. But perhaps you don't like logic...

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I'll try not to spark that debate in this thread. I don't understand high-level concepts in math, but thus far in my life, mathematical concepts that I don't understand have only given me frustration, I imagine due to how badly they were taught.

I couldn't understand most of the proofs I've seen for 1 = .999.... But I haven't understood a proof yet that I didn't see a fallacy in.

I'm going to be calm in this topic, so there's no need for personal insult.

Back to the topic, although, we were talking about mathematical concepts that have no relation to reality anyway...

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Yet in elementary school, I was so good at it, me and three or four other students in my school got to start learning basic algebra before everyone else. Once I got to high school, I'm not certain how, but it became different. It was no longer interesting or fun as it became increasingly, nonsensically complex.

If you've been paying attention to the 1 = .9999... repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.

Such nonsensical complexity is required to do physics. No nonsensically complex mathematics, no physics.

I sense that you are not comfortable with the abstract nature of mathematics. Hey, forget Modern. Look at Isaac Newton's analysis of the tides. Or, maybe not. It might cause your head to explode. Think about it. Mathematical operations on quantities smaller than you can imagine and yet they are not zero (infinitesimals). You are on the same page as Bishop Berkeley who gave a rather sophisticated refutation of the notion of infinitesimal (see -The Analyst- by George Berkeley). He was so on point that embarrassed mathematicians had to invent the concept of the limit (the major accomplishment of 19th century mathematics). Infinitesimals did not make a respectable comeback until 1960 with the work of Abraham Robinson on hyper-reals and non-Archimedian fields.

It is all nonsensical complexity, which is why you have a computer to complain on and with.

Bob Kolker

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Right, but my concern comes from the fact that people think these things *are* physical, actually part of the universe, and that we are simply incapable of perceiving or understanding them. If these concepts were more easily relateable to the real world, or put in the proper context, these confusions wouldn't occur.

The imaginary number i is the abstract mathematical idealization of a 90 degree (pi/2) counter clockwise turn of a unit length rigid rod. In a word, the imaginary unit is a rotation which is not all that removed from the real world. A proper understanding of complex numbers goes back at least 150 years. Even further if you include Euler's remarkable formula (I consider this the most beautiful equation in mathematics).

Behold!

exp (i*pi) = -1.

Thinking about it gives me goose bumps and makes the hair on the back of my neck stand up.

Bob Kolker

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I hate math. I honestly hate it. I have ever since high school. I only know what I have to know to be a practical programmer.

Do you know what a computable functions is? As a practical programmer you should.

You are going to love the Ackermann Function which is computable and not all that difficult to program using any of the standard high level languages:

Here it is:

A(0, n) = n + 1

A(m, 0) = A(m-1, 1) if m > 0

A(m, n) = A((m-1, A(m, n-1)) if m > 0 and n > 0

This simple looking function encapsulates succession (adding 1), addition, multiplication, exponentiation, superexponentiation and an infinite (unllmited) sequence of operations beyond that. Super-exponentiation is the application of exponentiation repeatedly. For example,

a^(a^(a^(.....))) n times.

the value for A(4,2) has nearly 20,000 digits in the decimal system and anyone who claims to have computed anything beyond A(4,4) is probably a fibber.

Since you are a practical programmer you might want to fiddle with this function. It can be programmed by by iteration and recursion. This function is used as an exercise in computer science courses.

For details see

http://en.wikipedia.org/wiki/Ackermann_function

Bob Kolker

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Back to the topic, although, we were talking about mathematical concepts that have no relation to reality anyway...

Almost every mathematical subject written on in the journals has a connection to physical reality. Sometimes the connection is long and abstract consisting of abstraction of abstraction (for example Category Theory). However all mathematics connects historically to answering two questions:

How many?

How much?

The most abstruse differential equation or topological theorem can be traced back to these questions, sometimes along a long and difficult path.

Some mathematics connects to visual arts, such a group theory which connects to symmetry, and projection. Any picture with a vanishing point has a connection to Projective Geometry, which is the geometry of incidence, rather than distance.

Bob Kolker

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Here's how I think of exponents.

Addition is adding numbers together. Easy to validate to sense perception.

2 + 4 = 6.

Multiplication is a "step up" from addition. Mutliplying a number is just adding it to itself a bunch of times.

2 * 4 = 2 + 2 + 2 + 2 = 8

I think of exponents as a "step up" from that.

24 = 2 * 2 * 2 * 2 = 16

Are you seriously saying that they didn't teach it to students this way?

Please go back and re-read the discussion, because you've completely skipped over my point. The examples you mention are perfectly fine - and they're exactly what I've said multiple times now (which you've clearly skipped over) -, but it all breaks down as soon as the exponent drops below 1, or goes negative, so I'm saying it needs to be explained to kids in a better way that takes the full context of the possibilities of exponents into account from the beginning.

How old are you?

Three years older than you, with a degree in physics. Great job confusing the intent of the discussion. Read the rest of your *own* post and you'll understand the true intent. People come to hate math - and I came to hate QM - because it is not explained in a proper conceptual way. Finding that proper conceptualization is the point of this discussion.

Edited by brian0918
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Ah, you're right. I did kinda miss that point. I don't know how to deal with exponents below 1, so I couldn't say anything about that.

I asked how old you were, because if you were younger, it would mean that you started school later than me, and that I could assume schools have gotten worse. I meant no insult.

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From Pisaturo

1 Introduction

In Introduction to Objectivist Epistemology (ITOE), Ayn Rand writes:

Since axiomatic concepts are not formed by differentiating one group of existents from others, but represent an integration of all existents, they have no Conceptual Common Denominator with anything else. They have no contraries, no alternatives. The contrary of the concept “table”—a non-table—is every other kind of existents. The contrary of the concept “man”—a non-man—is every other kind of existents. “Existence,” “identity” and “consciousness” have no contraries—only a void.[1]

In this essay, I will argue that the axiomatic concept of “existence” does indeed have Conceptual Common Denominators (CCDs) and, not literally “contraries,” but rather what I will call complements. I will also show that identifying the CCDs and complements sheds important light on this axiomatic concept. (An investigation into the CCDs and complements of the axiomatic concepts of consciousness and identity is outside the scope of this essay, except to the extent that it relates to the present analysis of “existence.”)[2]

Hmm...

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Whoa, I didn't see all those posts Robert made.

I'm actually just an aspiring web programmer so far with dreams of making video games someday. I have a personal quote: "I'm a programmer, not a mathematician." I reason out what math I need my program to perform, and only know the math I need to to do that. I don't need all that excess mathematical stuff that pure mathematicians learn.

I think I'm gonna duck out of intellectual topics for now though. I've realized that I have some problems with my thinking that I need to work out.

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So how is it that negative 5 multiplied by negative 5 equals positive 25?

Is that?: (-5) + (-5) + (-5) + (-5) + (-5) = 25

I would like to head to Las Vegas with this information. :)

Negative addition is subtraction.

- (-5) - (-5) - (-5) - (-5) - (-5) = 25

The only tricky part to that was remembering to also apply the subtraction operator to the first number.

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I think the point he's making is that, if multiplication is repeated addition, how do you multiply two negative numbers? If 5 x 5 is five plus itself five times, and 5 x -5 is -5 plus itself five times, then what is -5 x -5? -5 plus itself -5 times? How do you have -5 instances of anything? More problematically, what is 5 x pi? How do you have 3.14... infinite non-repeating decimal expansion instances of a thing? Or multiplication of complex numbers, and so on.

Turns out, when you study purer maths, particularly algebra, that we're not really interested in numbers in the first place. What we wanted all along, in the roll of multiplication, was just some operation on some set that satisfied certain basic properties (like closure, associativity, commutativity, having a zero-element for one of the operations [+], and a distinct 1-element for the other [x], and so on). Multiplication in the sense of repeated addition satisfied these properties, but then when we generalize mathematics, we keep the properties and let go of this common-sense way of understanding the operation.

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we keep the properties and let go of this common-sense way of understanding the operation.

In the process, you let go of all conceptual understanding. So what is the best way to understand -5*-5? You can make an appeal to generalization, but you'll have to show why the generalization is necessary, valid, appropriate, etc. Even then, that doesn't serve to help explain it more clearly.

There needs to be a better way of explaining the need or desire to generalize, or of conceptualizing it. Too often teachers simply say, "it turns out..." That leads to a mental dead end, and turns reason into faith.

On that note, there are a few good conceptual explanations here: http://www.mathsisfun.com/multiplying-negatives.html

Edited by brian0918
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Only the part of the concept that actually doesn't matter to what you're studying. Point is, not all mathematics is meant to count or measure. Complex numbers don't do this--they, among other things, chart directed line segments. It turns out the way it turns out, because the properties I mentioned above are useful in the situations they're useful in. For example, applications in number theory and questions like, "Given a tetrahedron, count the number of ways of coloring each face black or white, such that no two instances of the counting could be viewed as the same tetrahedron considered from different perspectives. (E.g. you could color the whole thing white except one face, and you could color the whole thing white except some other face, but they'll look like exactly the same coloring if in both cases you look at it from the angle of the black face. However, coloring it black on one side cannot be made to look the same as coloring it black on two sides.)" If you replace your domain of numbers with sides of a tetrahedron, and just stick to the requirements of closure, associativity, and so on, you can answer this question by means of an algebra.

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I think the point he's making is that, if multiplication is repeated addition, how do you multiply two negative numbers? If 5 x 5 is five plus itself five times, and 5 x -5 is -5 plus itself five times, then what is -5 x -5? -5 plus itself -5 times? How do you have -5 instances of anything?

It means do the opposite of addition five times. 5 x -5 means subtract 5, five times. -5 x -5 means subtract -5 five times.

Honestly if one is going to complain about numbers and math not corresponding to anything "in reality" the problem starts with negative numbers and even zero, not with pi.

The fact of the matter is, electrical engineering (your electrical service) and electronic engineering (your TV, CD/DVD/Blu Ray players, computers, etc.) would never have been developed without the use of complex numbers in the engineering--and those are some "real" number plus an "imaginary" number (dreadful names). Other fields of endeavor use groups of numbers (even groups of complex numbers) laid out in a rectangular pattern, and treat that whole thing as one quantity! It's all abstract... but if you think about it even the number "1" is an abstraction--it's up to you to find out something it ties to in reality.

[For the curious, "Real" numbers are the ones that appear on a number line--negative and positive integers, fractions, and "irrational" numbers like pi and the square root of two. "Imaginary" numbers are any of these, multiplied by the square root of minus one, which is written i. So an example of a complex number would be 3 + 2i. You can do just about anything with imaginary or even complex numbers you can do with real numbers, even use them in exponents. For example Gauss proved that eπi + 1 = 0. Yes, that's an irrational real number (2.718281828...), raised to an irrational "imaginary" number, equalling simply -1. Amusing that the five most important numbers in mathematics are all tied together that elegantly, no?]

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[For the curious, "Real" numbers are the ones that appear on a number line--negative and positive integers, fractions, and "irrational" numbers like pi and the square root of two. "Imaginary" numbers are any of these, multiplied by the square root of minus one, which is written i. So an example of a complex number would be 3 + 2i. You can do just about anything with imaginary or even complex numbers you can do with real numbers, even use them in exponents. For example Gauss proved that eπi + 1 = 0. Yes, that's an irrational real number (2.718281828...), raised to an irrational "imaginary" number, equalling simply -1. Amusing that the five most important numbers in mathematics are all tied together that elegantly, no?]

Seriously, we're not children here. I'm fairly certain most people in this discussion have seen that equation by now. Get in line behind everyone else who confuses a discussion on the philosophy of mathematics with a discussion on the science of mathematics.

It's strange, but I don't recall anyone having this problem in discussions about incoherent aspects of relativity, or QM. Why is it considered necessary to delve into the conceptual understanding underlying science, but not considered necessary for mathematics?

If this were a QM discussion, and I said things like, "trust me, the scientists understand it completely", or "at least it's no less incoherent than spacetime curvature in relativity" - both effectively arguments made by aleph_0 in this discussion - I would be considered dishonest or appealing to authority. Why is it acceptable in discussing the philosophy of mathematics?

Edited by brian0918
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