Is math truly objective?

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?

Damn double posts…sorry

So that’s the explanation…Heinz actually bottles ketchup in Klein bottles (presumably with a dimensionality of 57?).

Nothing is strictly objective. Everything depends on basic assumptions and axioms that are subjectively generated. But mathematics comes as close as humanity can to being truly objective.


Volume is measure in CUBIC units, surface area in SQUARE units. (Facetious) question:

Which is greater, one cubic foot or one square foot?

Squid: I hate to nitpick, but that riddle isn’t so much a riddle about math as a matter of trust.

Oh, yes, trust. People listening to the riddle assume that you will give them correct information, when in fact you have lied (they didn’t pay $27 at all). The effect is so powerful that even though I have heard the riddle many times (the first time when I was in grade 6 … 22 years ago, the same year I started karate amusingly) I still had to think about it for a second.

“Glitch … Window, large icons.” - Bob the Guardian

If the side of your cube is less than 6 units then the volume is less than the surface area. This is objective. I get the same results that you got. How is this subjective? (Also, you are comparing units squared to units cubed.)

Squid Really, that old riddle is about mixing money given with money received. It has an answer and does not show math is subjective. It only shows you must be careful with your signs.

Polycarp Good one. LOL

Virtually yours,



"Volume is measure in CUBIC units, surface area in SQUARE units. (Facetious) question:

Which is greater, one cubic foot or one square foot?"

Your right, but I wasn’t comparing 3D measurement with 2D measurement. Rather, the ratio of square to cubic units. The math problem changes with how you percieve the world, not as the world actually is. That’s my point.

Occam I think you are in fact comparing 2D units with 3D units when you compute the ratios. The units of measure are arbitrary, but once they are established, the results are objective, not subjective. That is, if we agree that the side of a cube is two units, it is an objective fact that we will both obtain the same value (8 square units) for the volume. I don’t see where subjectivity comes in at all really.

Virtually yours,


I hope I remember this correctly,

So you see, everything is derived from something.

If you want to talk real Math, not arithmetic, study Category Theory. If you can stay sane you will discover that even set theory, once considered the underpinning of number theory, has an underpinning. The more abstract the math is, the more difficult it is to conceptualize. Information Theory is another very interesting subject if you are interested in measuring intangibles. For those of us who love Math, everything else is just light and shadows.

I remember a story I read some time ago.

Five men are traveling on a train through England. Among the five are: a mathematician, a scientist, a doctor, a politician, and a layman.

As they travel through the country, the layman sees a black sheep and remarks, “Look! All sheep in England are black!”
The politician sees the black sheep and corrects the layman, “No, some sheep in England are black.”
The doctor comes in with, “Actually we should say, ‘At least one sheep in England is black.’”
Then the scientist says, “Well, at least one sheep in England is black on one side.”
The mathematician finally says, “At least one sheep in England is black on one side for a period of time.”

I love math too!

There’s always another beer.

Indeed - that’s why I opted for something easy, like graph theory! :slight_smile:

I’m stumped here. Why didn’t they pay $27?

I’m aware of the answer to the riddle–I just don’t understand why you’d say that they didn’t pay $27.

I don’t see anything in that version that is a “lie”, either. I see one misleading step, but it’s not a lie.

It’s like the old kids’ game: “If you have five apples and I take two, how many do I have?”–implying that you have to do arithmetic to get the answer, but there’s no “lie”.



Good points. The three did pay $27, of course, at $9 each. And it disbursed as $25 to the hotel and a $2 tip to the bellhop. And good analogy with the kids’ game.

You’re right, Mentock. The “lie” is the validity of adding $27 and $2. If you say it with the utmost seriousness it can take a person hours to realize that adding $27 and $2 doesn’t mean anything, but they assume that it is valid because you said so. I polled a dozen people or so (hardly a conclusive study) after asking them this riddle, and 9 said variations of “they assumed $27 + $2 was perfectly valid for the problem”. I.e. they trusted me not to mislead them with the facts of the problem.

Also, I like to look at the solution as:

they paid $25 for the room
they have $3
they gave the bellboy a $2 “tip”

hence, the confusing and incorrect, orgin of my original statement.

Most of the posts above follow the conventional common sense view of the issue. (4 objects are 4 objects, period)

This view neglects issues dealth with by Bertrand Russel and others about actual inconsistencies in math. In a long proof, he finally came up with 1=0. “Obviously” this is not true, even tho he proved it. What is wrong, either the idea that 1=0 OR one of the steps in his proof, all of which were sound mathematically. (I can try to find this and print it, although it is several hundred pages long - it is well known to those in the field).

An article in Scientific American last year challenged the idea that numbers can be infinite.

I think math is MORE objective, but from a philosophical standpoint, may not be ultimately objective - for the simple reason that it is a man made system (I am taking a position that no communication medium can be 100% objective, as all such systems are integrally tied to their creators and users).

Even if it describes qualities that are “out there”, some of the characterizations and how they are re-applied may be man made.

Positive and negative numbers are a good example. We could have created neutral numbers as well, and created a rule for them as such - 2 positives = a neutral, 2 negatives = a neutral, and 2 neutrals = a neutral.

An organism with tri-lateral symetery might do the same. Human math is very concerned with dualist divisions - thus the binary computer. Three states could be used to encode information.

We also confuse zero, the place holder with an actual number, having qualities (such as: it is even). The ancients did not think this way, thus 1BC is followed by 1AD. Zero used as an actual number yields non-unique counter-intuitive results, results I often felt were glossed over. (0 x anything = 0, this only occurs with zero. Dividing by zero seems more appropriate to Buddhism than mathematics).

While most of the posts have followed the common sense approach that it is objective - this is by no means universal in philosophy these days.

I had considered this a problem previously. You need to apply Occam’s razor Occam.

To “convert” the units from inches to feet, compute 1 square foot as 12x12 inches. One cubed foot the same equalling 1728 cubic inches. Convert you units and the ratios agree.

There is no rule in math that the ratio of a cube’s surface area to volume is a constant. The ratio is a function of the number of units dividing the side of the cube, relative to the same cube measured as a unit of 1.


Maybe we ought to look this up?


Rob - I suppose you could invent a system with positive, negative, and neutral numbers. I’m not sure how it would be made consistent, or what you would do with it, but it would hardly be the first trinary system used in important mathematics. For instance, the Cantor set and Cantor function, known to every grad student in math, are derived in base 3.

I’d be interested in the month and year on that Scientific American article about infinite numbers. I looked through their website (link above) at titles of articles going back to Jan. '98, and didn’t find anything that appeared relevant.

Maybe he was thinking about the April article about finite space.

If you want to take the position that math is purely subjective, I wouldn’t argue that either. It depends on what you mean by subjective, math is not related to the material world, it is a human construct that sometimes has applications in the physical world. There are plenty of mathematical branches that have no know application to the material world, they are pure abstractions. We concentrate our attention an research on the math that has a payback, but that changes over time. Computer scientists use mathematical constructs that were discovered before computers existed. Once the computer existed there was an application for the math, but before that constructions like finite state machines and Turing machines were used to examine the nature of information and information processing as an abstraction.

One of the beauties of math is that it is unchanging, it doesn’t depend on attributes of the material world so it isn’t modified when new discoveries are made. This is in disctinction to physics, which applies math to describe the world. Physics must adapt to new information, math doesn’t have to.

RobRoy said:

                    An organism with tri-lateral symetery might do the same. Human math is very concerned with dualist divisions - thus the binary computer. Three states could be used to encode information.

Some branches of math, in particular boolean algebras and their isomorphisms, are based on 2 values but there are other algebras, Heyting algebras for example, that are not.

One of the “articles of faith” in Computer Science is the Church-Turing thesis, which states : A number-theoretic function is computable by an effective procedure if and only if it is Turing computable.

Now since “Turing computable” is well defined and mathematically precise it falls within the world of math. An effective procedure is a concept from the real world, so it is impossible to “prove” the C-T thesis (proofs really only apply within the world of math, though the term is used very sloppily in general usage), but it has not been shown wrong yet and it was proposed in 1936. What this says is that any procedure or information which can be codified can also be represented in a Turing machine. And Turing machines only need 0 and 1 to represent data.