And assuming you are not using base 2 or base 3 math

Sure, but those are fallacies, most of them fairly easy to work out. A slightly trickier one than that, namely that: "Because 4>2 therefore 2>4" was thrown at us in my very first maths tut. The first time the prof actually paid me any attention was when he looked over my sholder and saw that I had cracked it. He was disgusted to hear that I was studying entomology instead of maths and was not officially a member of the class.

But in this thread most of the folk are speaking of operations that really are valid in particular circumstances. In most everyday terms that is a more treacherous problem than most of the deliberate fallacies.

I am answering the comments in the branch at this level because it gets hard to follow the indented posts.

Jon, you produced an invalid result, so I assumed you performed an illegal operation.

Was it as simple as:

(a) Multiple both sides of the inequality 4>2 by -1.

(b) Add 6 to each side.

Or do you have a more obfuscated version (how dare I ask!).

I think this discussion goes right back to Aristotle (or another Greek of similar vintage). The question is pretty much:

Three clouds; three pebbles; three goats; three thoughts; three olives; when you take away clouds, pebbles, goats, thoughts, olives, then what do you have left? The concept of threeness! Each such ....ness is an integer, and there is a reasonably obvious rule to move between such concepts. This rule permits of repetition, and thus establishes the countable numbers.

These Greek guys never really accepted the concept of negative integers, which is odd as they lived around 300 BC and were therefore a product of negative history.

I seem to remember a recent thread that berated mathematicians for permitting irrational and transcendental numbers to exist. That was the thin end of the wedge (a mathematical wedge measuring 0 units at one end, and 1 unit at the other, and therefore of indeterminate thickness anywhere between.) We should have foreseen this fundamentalist attack on these integers. Ah well, back to One, Two, Many.

Hi Chris,

>Jon, you produced an invalid result, so I assumed you performed an illegal operation.<

Necessarily!  :-)

>Was it as simple as:

>(a) Multiple both sides of the inequality 4>2 by -1.

>(b) Add 6 to each side.

>Or do you have a more obfuscated version (how dare I ask!).

Wellll...  Just a weeenie bit! My perpetration includes logs, which are higher-mathematics-and-therefore-beyond-the-scope-of... 

Let's see what I can reconstruct from the dark backward and abysm of time and memory:

4>2  (Granted? Unless there are four sparrow eggs and two goose eggs? <grmph!> )

Therefore log(4)>log(2) (log is a monotonic function, right?)

Therefore 2 log(2)>log(2) (log is a monotonic function, and 4=2^2, right?)

Therefore 2 log (1/2)<log(1/2) (log is a monotonic function... etc.)

Therefore 4 log(1/2)<2log(1/2) (Multipplliccationntion, right?)

Therefore 4 < 2 (divide by log (1/2) Log (1/2) being a constant, right?)

           QED = mc^2 right?

Mmmmm...food for thought, i know that when a receipe says 4 eggs it confuses me (even as a chef), because all eggs are different sizes and will effect the outcome, so if you use 4 large eggs you could really being using the fluid amount of 5 eggs, but i am sure thats not what you mean but thought i would just put it on the table :)

Or, put another way, why do mathematicians persist in applying axioms to real life?

Hello Delicate,

could You give an example for this?

Georg

Here I go along with Georg.

You say: "For a cook, 2 apples + 2 apples might well accurately equal 5 apples if those 4 apples are larger than normal."   

Wrong, as I said to Nicholas.

You are speaking of elements of two sets such that in enumerating four elements of one set ("large" apples"), and five elements of another set ("smaller" apples), you take the same quantity of material. This is a mathematically valid class of operations, but has nothing to do with integer arithmetic, where 2+2=4 and nothing but.  

You say: "The mathematician would argue that 2 large apples + 2 large apples must equal 4 large apples. Correct. That’s the mathematical axiom Jon Richfield is talking about."

Wrong. the theorem that arises out of the axioms is 2+2=4. It says nothing about apples, mugs, flashes of light or any other quantification, though it is useful in enumeration among other operations. It has nothing to do with how big your pie might be, just how many times you must take  an apple if you want four items.  If anyone wants you to fetch enough apples for a pie and he does not know how large the apples are, then he is innumerate if he tells you how many to bring. He should instead tell you what mass to bring.

You say: >The trouble is, in reality no apple is the same size as another, so the mathematician’s axiom is limited somewhat to arithmetic theory.<

Decidedly! No axiom is relevant to any theory that does not include it. What made you think that any mathematician thought anything else? Suppose I asked you how many metres in a cubic foot? Would you say that metres are no good because there are none in a cubic foot, so that they are useless when one wants volume rather than length or mass? Or if I asked how many atoms in a kiss? Or gaols in a goal? If you talk about the wrong things for the wrong words, you find that ordinary commonsense words stop making common sense.  

>So why should mathematicians get the final say?<

No mathematician is interested in the final say unless you make invalid assertions in mathematical contexts. How bothered would you be if someone said that glue was faster than lightning? You might argue briefly, but as soon you found that he was not interested in talking sense about anything of interest to  you (or anything else probably) , you would seek another conversation. Similarly a mathematician that had to argue whether 2+2 meant different things for big apples and small, would prefer to discuss something more interesting.  

>The cook’s application is commonsensical and thus more accurate and fair, so in real life 2 + 2 doesn’t always equal 4. <

In real life, 2+2 = 4 precisely. Always. Count the apple cores if you don't believe me.  That is real common sense.

And being more accurate neither means more fair nor less unless it makes sense in context. If I gave you a share of the apple pie half as large as everyone else's, would you feel it was any fairer just because I measured your half-share with a micrometer? 

>Using the equation 2 + 2 = 5, the apple pie turns out normally, as intended. Nothing meaningless about that.<

Totally meaningless; you did not say how much apple is needed. If the pie turned out as intended, then it is because the pie required say, half a kilo of apple slices, and that was what four 275-gramme apples yielded, instead of five 230 gram apples.The pigs would prefer the latter of course, because it means more peel and core for them. Simple quantity surveying!

Right?

And if you want it to have meaning, then 2+2=4.

Still right?

>Or, put another way, why do mathematicians persist in applying axioms to real life?<

They don't. Most mathematical axioms have nothing to do with anything but abstract maths.  It is the non-mathematician who mis-applies axioms in real life, and feels aggrieved because maths makes no sense.  

Right?  :-)

I don't know about 2 + 2 = 5, so let's assume that it is true then 2 + 2 = 4 is not true? 2 means just that, 2 nothing less, nothing more, if you want more or less it should be shown, for example baking a cake, takes in part 2.5 cups of flower, we can round it up to 3 but the results will, well not be true to your taste buds.

To use 2 + 2 = 5 in any scenario the reader can interpret it to be anything, 1.5 rounded up to 2 or 2.4 rounded down to 2, which is which, so my conclusion is: what you write should reflect the exact meaning to be interpreted and used correctly.