Is mathematics truly objective and universal, or is it just a human thing?
Is it merely human to count things?
What if there is a system of true objectivity in this universe and we don’t even freakin’ know it yet?
There’s always another beer.
I mean, perhaps if we can question the axioms, we can dig deeper into our psychology and the true meaning of things.
Life is interesting in the drunken moments.
There’s always another beer.
Math is about as objective as you can get.
If I put four objects in a row and ask you how many there are, you will (presuming you can count) say “four,” as will everybody else (who can count) in the world. There is absolutely no subjectivity involved. Now, if I asked you if those objects were pretty, smooth, or even what color they were, subjectivity would come up.
Beeruser:
What great questions! “Computers are useless. They only give you answers.” — Pablo Picasso
In my opinion, there is evidence that mathematics is truly objective and universal. There seem to be myriad practical applications of mathematics throughout all the sciences and scientific disciplines.
I honestly don’t believe that science, as we know it, would even be possible without mathematics. For one thing, the scientific method, as I understand it, does not let us just qualify; it requires that we quantify. If science is objective, then I think math must be because numbers are the very expression of its measurements.
The reverse is also true, I think. If math is not objective, then neither can science be objective for pretty obvious reasons. The measurements you make would be fairly meaningless. That is, they wouldn’t be at all useful, because if the measurements that confirm laws governing electromagnetism apply for you but not for me, then we are living in two different universes anyway.
It might be. I’ve seen at least one mathematician who hypothesized that counting began as a tool to document property. Man counted his possessions, which enable him to account for his property and determine whether any of it is missing, or (a bit later) enable him to discern progress (or profit). In other words, her theory is that counting was driven by an economic necessity.
But it might be important to remember that counting is only one attribute of numbers. Besides cardinality, there is also ordinality. And it is possible to define a manifold number system, like the so-called surreal numbers. They are not only cardinal and ordinal, they are hierarchical.
I think the implications of that would be almost unimaginable. I think that it is safe to say, at the very least, that we would have to start all over.
Mathematics itself is inherently objective. In math, you define all of your axioms and then prove theorems, etc., and anything you do under that system is correct, according to the axioms you laid out. The opportunity for being something other than objective comes when you try to link mathematics to the physical world. For example, if you wanted to use mathematical formulae to plot the path of a rocket, you had better make sure that the geometry system you are using is a close match to the physical world. So, mathematics is inherently objective, but the application is not necessarily objective, although most people make every effort to make objective use of mathematics.
I’d say mathematics is objective, given the choice of axioms. Take Euclidean geometry, for instance. It begins with five postulates, including the infamous ‘parallel postulate’ that decrees that if you have a line L and a point P, there’s only one line through P parallel to L.
If you replace that postulate with (a) one that says there are no parallels through P, or (b) one that says there are two such parallels, you get entirely new geometries. What choice one makes regarding postulates is subjective. From that point, what’s provable is set; mathematicians just discover what is already true.
Math is objective. But too often it is mis-applied to give subjective information, without acknowledging the inherent biases in doing so.
The sun is 93,000,000 (give or take a few) miles from earth. Is this near or far? For a human standing on earth 2,000 years ago, it is incomprehensibly far. For someone 2,000 years in the future it may be like walking 20 feet next door.
Offer a 2 year his choice of 10 pennies or 1 quarter, he’ll pick the pennies. By 6, most kids are translating those coins into how much candy they’ll buy, and would pick the 1 quarter.
Math as a science in itself is as purely objective as anything we can comprehend. It’s also pretty useless without applying it to real life situations. But as many a student has asked when faced with a train speed problem, “But why would I want to go to Omaha?”
Sue from El Paso
Experience is what you get when you didn’t get what you wanted.
Rufus, does Godel’s Incompleteness Theorem have any bearing on mathematical objectivity?
After all, the axioms are supplied not by mathematics, but by the mathematician.
Or am I blowing smoke here?
When the pin is pulled, Mr.Grenade is no longer our friend.
Thanks for the great post, Libertarian. I always felt that all math came from geometry and our ability to count. What would the universe look like to us if there was no economic scarcity and thus eliminating our need to count?
–RTFirefly
Are you saying math is not a human invention? This could be an interesting debate. Could the principles of mathematics exist aside from us who have done nothing but discover them?
There’s always another beer.
Math is the epitome of objective truth. Only a poststructuralist would dare to argue that the square root of 9 is three to you and might be something different (with equal validity) to someone else.
Wally: the Gödelian theorem states that there are true statements that can be made within the system of mathematics itself which cannot be proven by any set of other statements that can be made within that same system. It has profound implications for how we think of objectivity itself (i.e., some things may be true yet impossible to derive from within the system in which they are truths), but it in no way implies that truths that can be derived through mathematical deduction are of dubious reliability within mathematics.
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To play the devil’s advocate, here’s an argument to the “subjectivity” of math.
What is the square root of (-4)? You know it is an imaginary number with no “practical” meaning, right? Yet, a big chunk of our physical world is only explained by using imagnary numbers!
I say this means there are “holes” in math in which we are asked to bend the “rules” of what we hold as true - to make the model fit what we observe. Thus, I’d conclude there is some subjectivity in higher mathematics.
I would like to get my hands on the person who came up with the term “imaginary” numbers.
The imaginary doesn’t imply that there is no practical meaning. They are as you point out, quite useful. The imaginary simply points out that I cannot show you an “imaginary number” amount of apples, but they are still numbers that exist as part of the underlying nature of the universe. There is no doubt about this since they were quite useful in solving many problems, particularly in the physics field.
Glitch:
You think it’s the same guy who designed the ketchup bottle? 
I hope they arent’ really teaching in schools these days that complex numbers have no practical meaning, though I wouldn’t doubt it. Complex numbers are indispensible for modelling systems with dual components, like electromagnetism (electricity and magnetism). At least, as I see it.
Whenever the “imaginary numbers” thingie comes up, I show folks a 2-D graph with x,y coordinates and explain how useful it is to be able to march off at right angles to the linear stream of “less” and “more” that we think of as rational numbers. I explain that it is mathematically more efficient to express each point as a single numeric string than defining points with separate x and y coordinates. With the emphasis on utilitarian usefulness rather than disconnected mathematical theory, they don’t glaze over.
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Yes, complex numbers have practical application, but it’s like a “back door” to math. (Or, a loophole for mathematicians?). I guess if you can conceptualize higher math theory, then it is intuitive. For me, I reached a point of “don’t ask, don’t tell” - just accept it! (I just needed to apply it, not derive it from scratch everytime!)
I can apply the math, but it might as well be black magic about just why certain functions, transformations, and relationships work on the plane of complex numbers. I only suggest that things in higher mathematics reach a point where people, like LaPlace, just made up their own rules to model what they saw.
Jinx said:
Yeah, I think the hardest part in learning math is when you come to a point where you can’t visualise or relate to the concepts anymore. For a lot of people, this seems to be imaginary numbers or calculus (for me it was group theory, ugh). There’s just some point when you have to switch from trying to relate the concepts to familiar experiences and switch over to just playing by the rules.
I would say yes, for our purposes, math is objective and universal. But as BeerUser pointed out, it may be simply human. How would we know, as we are simply human?
If there were to be a true universal objective application, and we were to discover it, perhaps we would have to start all over. Perhaps not. I work as an accountant, the company I work for recently completely changed software applications. They found a better mouse trap. Is it inconceivable to think that the course of humanity may alter by such a discovery? I should hope not. Let the dreamers dream, let the inventors invent. You never know what lies ahead.
Always be ready to speak your mind and a base man will avoid you.
-William Blake
Beeruser You mentioned questioning the axioms. Very good. Non-euclidian(sp?) geometry was discovered by doing just that. It does not mean that math is not objective. But I think I see your point. The decision of which axioms to work with is not objective, but I think this falls under meta-mathematics. Mathematics is objective, but meta-mathematics is not objective.
Jinx The set of numbers is whatever “number” is defined to be. That is if you are considering only real numbers, then there is no number that is the square roor of -4. You don’t bend the rules, you replace them with another set of rules. If by “number” you mean complex then, there is a number(actually, two numbers) that equal the square root of -4.
As far as no “practical” application. I think you are wrong, but I really don’t care. I enjoy math because it’s fun, not because it does or does not practical applications.
Virtually yours,
DrMatrix
Take a cube that is one unit on each side.
The volume is 1 unit.
The surface area is 6 units.
Take a cube that is 8 units on each side.
The volume is 512.
The surface area is 384.
So…what you call a unit changes the reality of math. If a cube is one foot on each side the surface area is larger than volume. If you measure that exact same cube in inches the volume is larger than the surface area.
How more subjective can you get?
I think that mathmatics are objective only if the bounds of the system are setr objectively. Or at least rigidly. I agree with David that if you put 4 objects on the table, everyone will count 4 objects. But what if you were to ask how many objects are not on the table?
I remember a math riddle my High school algebra teacher used to explain perspective in math. 3 people get a room together at a hotel. The owner charges them $30. They each give him $10. Later the owner realizes that he charged too much. He gives the bellhop $5 to return to the people. The bellhop gives up trying to divide the $5 into 3 equal parts, so he gives each person $1 and pockets the remaining $2. Now each person has paid $9 for the room which totals $27, and the bellhop has $2 for a total of $29. What happened to the other dollar?