Mathematics doesn't provide universal truth with human perspective being irrelevant

This thread grows out of a recent thread in which we were discussing the nature of philosophical truth. Somebody claimed that mathematics provides a way to universal truths that are independent of human opinion. Another way I’ve heard it put, in a thread awhile back, is that in math everybody gets the same answer. I, personally, disagree with this statement and I provided a counterexample. Nobody responded, which is just as well given that it was tangential to the thread in question. Now I’d like to bring the topic up again in its own thread. I’d also like to mention that I was inspired by this excellent blog post.

My counterexample is this:

For those who don’t want to click, it’s a Calvin and Hobbes strip where Calvin’s dad is trying to teach him addition. The dialogue is this:

Dad: Here, maybe this will make more sense. I have eight pennies and I ask you for four more.
Calvin: I say forget it. You’re the one with a steady paycheck.
Dad: Just give me the four pennies. Good. How much money do I have now?
Calvin: Investments and all?
Dad: No. Just here on the table.
Calvin: Eight cents.
Dad: No, eight plus four is twelve. See? count them up.
Calvin [pointing]: But those four are mine!

Now when I first read this as a child, I assumed that the disagreement arose from Calvin being wrong while his Dad was right. Now, however, I realize that both of them can be right simultaneously. Calvin’s dad views this as an instance of addition taking place in the ring of integers. A set of eight pennies combines with a set of four pennies to become a set of twelve indistinguishable pennies. (8+4=12).

Calvin, on the other hand, views the operation as occuring in the non-abelian free group generated by the integers. He sees two different sets, his pennies and his Dad’s pennies. They combine into a single mathematical object, but remain distinct sets with different properties. (8+4=84)

Both Calvin and his Dad are entirely correct in their computations. They arrived at different answers because they assigned different meanings to the concept of “eight plus four”.

Of course one might respond by adjusting the claim slightly to say that everybody will get the same answer in math if they first meet and agree on the meaning of what is being said. That, however, basically takes the core out of the argument. Defenders of universal truth in mathematics would typically want to say that results in math don’t require any discussion of subjective, emotional issues such as how to interpret the pennies.

One might also respond that this issue is rather trivial. I don’t think that it’s trivial at all, but instead it’s at the heart of many major issues today. Many times in debates about government policy we hear people saying “You can use my money to _____________”. The thing is that some people view the money they pay in taxes as remaining their money while others view it as becoming the government’s money and no longer having any connection to the taxpayer that it came from. It’s really the same disagreement that Calvin and his Dad had, only with larger amounts of money.

I’m not sure where the subjectivity is here. The integers and the operation of addition on them are independent of human perspective. The free group on a countable basis and the ordinary group of integers under addition are both perfectly valid and well-defined mathematical objects; the disagreement is about which of them is a better model for the real-world situation of moving pennies into different piles. Pouring a cup of salt into a cup of water gives you a cup of salt water, but that doesn’t mean that 1 + 1 = 1; it just means that the integers are a poor model for the operation in that situation. More directly, if I decide to denote the group operation on the integers with * instead of + (as is standard for arbitrary, not necessarily abelian groups), the statement “1 * 2 = 3” doesn’t claim that centuries of mathematicians have been off by 1; it just means that my notation is a bit unusual.

I would respond with that; that’s the whole point. It sucks to have to use English to describe math, but part of the whole point of math is to define things precisely enough that if you ask 100 mathematicians what, say, the rank of the integers as a free group is, then you’ll get 100 answers of “one”. Your argument might be more compelling if it were talking about foundational issues about how to define a proper set theory and what the ultimate underlying assumptions of modern math are, but this is really just quibbling about English semantics.

But then, by that example it follows that one side is dishonest or clueless.

No matter how you cut it, at the end of the day, when he goes to the store Calvin would not get a chocolate bar, but he would be lucky to just get a free sample of chocolate in the store (if we can use the 8+4=84 mentioned).

Calvin is just complicating the lesson, that made it funny, but that was not the point of addition.

The issue is not addition, it is the meaning of “How much money do I have now?” Calvin considered his four pennies not to be included in this definition - where you got 84 from I don’t know. Not even in the strip. The axiomatic definition of arithmetic does not include possession.

You might as well make your point with two people using different bases. Or use the example from Annie Hall

Shrink to Annie: How often do you have sex?
Annie: Three times a week. All the time!

Shrink to Alvy: How often do you have sex?
Alvy: Three times a week. Hardly ever!

Of course, if Dad had said, “You have four pennies and I give you eight more, how many pennies do you have now?” Calvin would have answered, “Twelve” without reservation. That’s the point of the strip.

Math is universal-the English language, however, is an ungodly mess. Don’t blame the math when the English is at fault.

I agree that it’s important to debate whose money is being spent, and for what purposes. That is not a math debate.

As others have pointed out, the differences are not due to the different concepts of “eight plus four,” but due to differences in define "do I have" in the question “how much money do I have now?” “Have” can be Own or it can be Hold in My Hand. That’s not a math or a computation issue at all, but English language.

Bigger issue, why are you search for “universal truth” at all? Would you really feel better if this thread resulted in a consensus that “mathematics doesn’t provide universal truth?” Who cares?

I expect it’s something to do with his religion again, like his periodic threads arguing against evolution and so forth.

I think this is a bit of an “if a tree falls in a forest…” debate.

Humans (well, maybe some of the more intelligent animals as well) are the only creatures that can give numbers any meaning. In a world without intelligence, “12” is as absolutely meaningless as any other number. We are the ones, by interpreting and attaching significance to it, who make “12” mean “12.” How can “truth” have any meaning without someone to perceive it? Without people, what is just is, and doesn’t have or have the possibility of any description.

Math is a language. It is a way to describe things. It is language that is well suited to describing the world in a way that is useful because it is predictive. But the math itself is still just a metaphor. It doesn’t have any inherent meaning of it’s own.

:rolleyes: Og, “Bob” and Allah, is this another episode of “God-Man vs. Freshman Philosophy Major Man”?

I thought non-Euclidean geometry and Godel put an end to the notion that mathmatics are a universal truth?

There are several posts in this thread that point out the confusion of the concept of Mathematics and other psychological and social ideas or behaviors, resulting in a conclusion that seems paradoxical, but it’s just meaningless.

Math is the result of the capability of the human brain to express its own abstract ideas with more simplified abstract symbols, and of our need to try to devise deterministic rules that govern those symbols. This way we are trying to comprehend nature independently of human bias, something I called a universal truth in the previous threads.

The every day language we’re using trying to communicate social situations, as the economy example you mentioned, is severely inadequate to describe math and its rules. So, be careful when you use common language to refer to math and all its amazing glory.

An other related point:

Plato is commendable for coming up with his eternal quality of ideas, but the universe doesn’t know what the number 17 is and it doesn’t do addition either. Math is our invention to help us understand what’s going on around us. The only reason it has been so successful in doing so is that we have been able to disengage our self-absorbed ego from analyzing our observations or thoughts about how the world works.

From a math or universal perspective Calvin is wrong. It’s only the frailty of ego that makes Calvin disregard simple addition, and that has nothing to do with Mathematics.

No.

No, Godel simply means that mathematical systems can’t be both finite and complete. And non-euclidean geometry just means that mathematics can handle things that aren’t flat.

To respond to just the OP title:

Hammers don’t provide houses, either. A tool is a means to an end. Depending on the house’s design, the end changes, but the tool always works the same.

What did you think was excellent about that blog post? It says nothing.

In our society we use mathematics defined in a certain way to handle economic transactions. However, that’s not the only possible way. In this particular example Calvin is refusing to accept the mainstream, emotionally loaded approach to economic transactions that prevails in his society. He’s thinking outside the box. However, from a mathematical perspective he has as much claim to correctness as his father does. To say that he’s “dishonest or clueless” because he refuses to be boxed in by the narrow-minded approach to mathematics that’s popular in his society is merely conformist thinking.

Well you’d be wrong since this thread has nothing to do with religion and I’ve never made a thread arguing aganist evolution periodically or non-periodically.

I would think that your metaphor points in the opposite direction that you want it to. Anyone who’s actually done construction work can surely testify that there are many different styles and methods for using a hammer, that it can be used for many different tasks, and that one person’s understanding of the correct use of a hammer may be quite different from another’s.

This says nothing about mathematics, and merely points out vaguaries in the English language. It’s not that there are two different answers to the same question, it’s that two different questions have been asked, each with a correct mathematical answer.

That’s a totally incorrect characterization. That’s not what (Platonists & Formalists) would argue at all. You’ll want to familiarize yourself with their position because you’re debating a position they don’t have.

You are talking about “conventions” of communication between two people regarding interpretations of physical items (a “concrete” application of mathematics). That’s something else unrelated to their idea of mathematical transcendence.

The mathematical truth guys are contemplating at a much deeper level (the “abstract”) regarding the existence of “integers” and other mathematical objects. The “abstract” emphasis is unrelated to the messy conversion to the atoms that make up such things as “pennies”. The abstract also doesn’t require the atoms that make up the human brain itself to even “think” about pennies in the first place.

You are using analogies irrelevant to your argument. Is the concept of “love” a “universal truth” outside of human experience? Can immortal souls “love” each other? Can animals love each other? Can 2 single-cell micro-organisms love each other? I don’t know… but you can’t argue the point by saying that since roses wilt and divorces happen, it then proves that love is not transcendent. Those messy real world examples are not relevant to making the argument one way or the other.

(I say all the above even though I’m skeptical of the “universal truth of mathematics”.)