Good morning, Fednid48, Yazata, QQ, MD, Pete, Tach, everyone.

This is an interesting take on things. I guess I hadn't thought about the metaphysical status of infinitesimals. By definition, any infinitesimal is smaller than any arbitrary number we can choose to compare it with. So if a mathematical expression does contain an infinitesimal, throwing the infinitesimal away will always leave the value of the expression unchanged up to arbitrary precision. Since math is just a language for formalized logic anyway, I don't worry too much about whether throwing away the infinitesimal gives a correct description of what the number "really is"; insofar as the results are the same, I'm inclined to go with the simpler method, which is not to bother tracking infinitesimals. But it's interesting (and probably grants insight into some fundamental properties of arithmetic) that one can build math in a non-standard way using infinitesimals.

Your honest (as usual; Kudos!) self-observation allows that you had not before considered (at least not to the extent that I and some others here apparently have) the deeply important and comprehensively instructive "META-PHYSICAL status" infinitesimals. By implication, it is possible that you may also had not considered (again, at least not to the extent that I and some others here apparently have) the even more reality-pertinent 'REAL-PHYSICAL status" of the logically deducible and physically recognized "infinitesimal of effectiveness" which is inherent in the understandings and dynamical basis of QM "a 'something' tunneling through 'infinitesimal nothings' to 're-produce' that 'same something' elsewhere", as observed; of CHAOS THEORY "infinitesimal steps from starting simplicity towards infinite complexity", as observed; and, of FRACTAL MATHEMATICS "iteration equations/effects based on fractal infinitesimal variability between iterations", again, as observed.

So

*maybe we should*, as you put it,

**"...worry..."** about the infinitesimal in

*all* its contexts, Fednis48?

Especially if our common goal is 'completeness and cross-consistency' between

*all* abstract mathematical

*and* concrete physical 'systems of thought/modeling'?

Again, assuming our collective

*ultimate* aim IS to actually address THE reality and not just keep playing with NON-reality. I think humanity has grown up and left the basement of 'video games' and is now beginning to actually 'face the real world' outside the basement 'virtual world' disconnect with what's really important in the final analysis of what science is about.

Fednis48 to Yazata said:

That said, I'm a little disappointed that people seem receptive to Motor Daddy's arguments. Finer points of math philosophy aside, the idea that an object cannot be divided into three equal parts is *utter nonsense*. As gmilam alluded to back on page 2, the non-terminating decimal representation of 1/3 is just an artifact of the base-10 system we use. In a base-9 system, 1/3=0.3 (not repeating). Fundamentally, there is no reason why 1/3 is any less precise of a number than 1/2, or any other fraction. It might be a matter of mathematical formalism whether 1 and 0.(9) represent the same number of two numbers that are infinitesimally different from one another, but it's important to keep in mind that at most this is a question of *labeling*. Trying to extend this reasoning to argue against the reality of certain fractions is beyond inane.

Again, you seem to miss that the only counter-argument so far, against MD's REALITY-based approach to that 1/3 (ie, "divide something into 3 equal parts IN REALITY"-----whatever abstract number system you wish to play with in UN-reality) has been offered by Tach , using EXACTLY the same REALITY-based approach which he had 'derided' MD for using!

Here is Tach's supposed 'counter' example, effectively constituting a real 'division exercise' on a real thing (read MD's "pie" for Tach's "circle"):

Really? In 8-th grade they teach you how to inscribe an equilateral triangle in a circle. This means that you either haven't taken that class yet or that you flunked it.

I don't think that you understood the simple exercise. You are given a compass and a ruler. Draw a circle. Divide the circumference of the circle in 3 equal parts using the two tools given to you. You have 10 minutes from when this post is active. If you cannot do it , you flunked 8-th grade geometry. Live with it.

See, Fednis48? There is no REAL 'proof' of the a-priori/abstract

*assumption* by Tach (and now effectively repeated by yourself above) that any 'real division' scenario results in 3 'equal' parts instead of MD's perfectly valid 'real context' observation/argument (even using Tach's OWN example) that IN REALITY there is NO basis for making a-priori/abstract ASSUMPTION that the 'division' CAN give 3 'equal' parts AT ALL in that particular scenario.

Of course, IF we FIRST construct or 'compose', something shown to BE the SUM of PRIOR objects which INITIALLY SUPPLIED the 3 "equal' parts to some 'composite object' MADE from those 3 "equal' objects in the first instance, then it would be trivial to 'reverse' that operation and 'decompose' it into 3 equal parts. HOWEVER, since the 'pie/circle' in this particular context has NOT PREVIOUSLY BEEN 'composed' using 3 'equal' parts in the first place, THEN can be NO A-PRIORI/ABSTRACT ASSUMPTION that it can NOW, in reality, BE 'decomposed' into 3 "equal" parts. It cannot BE 'proven' in that case where no 'composition' case has been demonstrated to arrive AT the 'pie/circle' REAL OBJECT we want to 'divide' via 1/3 in REALITY, as per MD's point made so far.

**( * )**
Hence the math-versus-reality perspectives 'impasse'; and where the 'last step' infinitesimal-of-effectiveness 'non-zero difference' comes in to save the day; which would make MD's (and others, including myself) observe that always (whatever number system/abstraction one plays with), in reality, the expression 1/3 represents/results in:

*At least ONE of those three parts* being slightly (by an unavoidable infinitesimal of effectiveness) greater than the other TWO parts.
Anyhow, this discussion should highlight yet again the possible dangers of letting our 'mathematics' rule us blindly in all contexts, since it tends to depart from reality more and more as UN-real abstraction/assumption is piled upon UN-real abstraction/assumption.

Yes, Mathematics is useful, but let's not let it run away with itself and us, and so insidiously OBSCURING from our ken more and more that reality which we are striving to elucidate for REAL and not just for VIRTUAL. Yes?

Let's NOW actually "...worry about..." and really consider properly and exhaustively all the contextual aspects/effects/meanings etc OF that "infinitesimal of effectiveness' LAST REAL STEP between something and zero/balance/singularity etc etc states which occur in reality but which bamboozle our mathematics because, as axiomatically defined so far, the maths gives infinities and singularities when it breaks down and our current equations 'blows up' to indicate the end of its 'domain of applicability' boundary conditions which it cannot handle with any reality sense result.

Thanks for your time and trouble in making your own very interesting contribution to this

**PHILOSOPHY of MATHS** thread, Fednis48, Yazata, MD, everyone. Much appreciated; and by more than just Quantum Quack, I assure you!

Great thread, Quantum Quack; Kudos for starting it and setting just the right tone for polite and insightful discourse roght from the start! Cheers!

**( * )**This is a subtle but extremely important 'contextual reality' aspect requiring careful consideration to understand/discuss these things properly. For example, we can in the first instance 'compose' a '6' from three "equal" parts of '2'; and its reverse is trivially achieved by dividing precisely into its original three "equal parts of '2'. HOWEVER, we CANNOT DEMONSTRABLY IN REALITY INITIALLY 'compose' directly a REAL "UNIT WHOLE" object (a UNIT WHOLE 'pie', a UNIT WHOLE 'circle') which is NOT ALREADY amenable to being 'composed' FROM three "equal" parts in the first place. Unless anyone can INITIALLY MAKE a REAL, WHOLE UNIT 'pie' or 'circle' FROM THREE REAL WHOLE UNITS which can BE demonstrated in reality to BE "equal" to each other, then no amount of starting from the 'other end' can PROVABLY (not abstractly/assumedly) 'derive' three "equal" parts in reality (again, not abstract/assumed) division operation applied to those reality cases.