The Maffematics Thread

Discussion Open Forum
21

1+1 = 2?

Sorry if I intrude and sorry if what I say is obvious to you. :blush:

In mathematical terms, you can build as many “theories” as you like, i.e starting with the
premises of you own choosing, and that’s what people really do.
Then, merely by deduction, you can establish the properties (theorems) that logically follow.
1+1 = 2 may be understood in the context of the Integers set, i.e. [0, 1, 2, …, N] assorted with
what is called an internal law, that is, addition.

In that context, the only way you could not have 1+1 = 2 would be in “Base 2”.

The base is the number of basic symbols used to represent the set of numbers: in Base 10, you
have the well-know ten symbols, from 0 to 9. In Base 2, you have 2 symbols, 0 and 1.

Whereas in Base 10, 1+1 always makes 2, in Base 2, 1+1 makes “10”.

In Base 2, successive numbers would not be 0, 1, 2, etc. but 0, 1, 10, 11, 100, 101, 110, 111,
1000, etc. That is, you don’t have “2”, but simply because “2” is now written “10”.

Mathematics of course are not the physical world.

This is still the real world, but it all hangs on the way we understand it, rather than what it is.

Particular mathematical theories are applied to the real world essentially for their predictive
capabilities, not as a description of was the real world is. They merely represent our
understanding of it. They may be both successful at predicting real world results – i.e. a
rocket on Mars – and misleading in terms of describing the reality behind our perception of it.
The real world, assuming it exist at all, can be assumed to be only accessible, and most
probably not all of it, through our senses and through diverse measurement apparatus. All of
this is merely approximative and can usually be improved over time, thereby forcing people
to reconsider well-established theories.

However, in many other cases, as in the case of Newton, new theories offer new
understanding and new results, but often the new results are only marginally different from
the previous ones.

This certainly shows that the previous theory was wrong, but it also shows that you should
really expect that, sometime in the future, the new theory will also be proved wrong.

In the real word, 1+1 = 2 works fine, but there is no proof that by adding 1 discrete real object
to N identical real objects, you’ll end up with N+1 of the same objects, and that for any value
of N.

I’m not even sure if it’s not already the general situation in Quantum Physics.

EB

22

sure, in base 2, 1+1=0. But usually we do not write it like this. We use a subscript “2” to show it’s base 2 math. Too bad there’s no subscript mechanism or I could show you your error.

23

I have to say something because I have a graduate degree in Mathematics.
1+1=2 is true by definition
This is true for the real number base ten system that everyone uses. There is no proof.

I think it was Bertrand Russell who proved that it is not possible to prove everything. You have to start with assumptions, definitions, and in Mathematics–completely undefined terms.

24

twocs wrote
“sure, in base 2, 1+1=0. But usually we do not write it like
this. We use a subscript “2” to show it’s base 2 math. Too bad
there’s no subscript mechanism or I could show you your error.”

I don’t know who you are replying to, but if it’s me, I didn’t say that in Base 2 “1+1= 0”.

I said:
"the only way you could not have 1+1 = 2 would be in “Base 2”.

EB

25

[quote=“Scuba”]This reminds me of that interesting statment:
“I always tell lies” If you did always tell lies then you cannot use this statement to communicate the fact since this would be a true statement & thus invalidate your claim to always tell lies …[/quote]
This is graded logic…by telling a lie people tend to think that the entire statement is a paradox, but now that I’ve thought about it (whether anyone else has or if this is the point of the statement, this is my own independent thought), you can negate one thing and it could still work…People negate the “always” for the direct opposite of “never”…what if you negate it to “not always” so that if “I always tell lies” becomes “I do not always tell lies”…then it is probably the truth, because not only is the original statement a lie, the ‘truth’ is also a truth. I am not sure how it relates to mathematics, but dammit, I finally figured out that sentence all by myself.

26

Kirk: Everything Harry tells you is a lie.
Mudd: Listen to this carefully, Norman. I am lying.
Norman: You say you are lying, but if everything you say is a lie, then you are telling the truth, but … you cannot tell the truth because everything you say is a lie. You lie – You tell the truth – But you cannot – Illogical! Illogical! Please explain. Only humans can explain their behavior. Please explain.

– I, Mudd

27

[quote=“ImaniOU”][quote=“Scuba”]This reminds me of that interesting statment:
“I always tell lies” If you did always tell lies then you cannot use this statement to communicate the fact since this would be a true statement & thus invalidate your claim to always tell lies …[/quote]
This is graded logic…by telling a lie people tend to think that the entire statement is a paradox, but now that I’ve thought about it (whether anyone else has or if this is the point of the statement, this is my own independent thought), you can negate one thing and it could still work…People negate the “always” for the direct opposite of “never”…what if you negate it to “not always” so that if “I always tell lies” becomes “I do not always tell lies”…then it is probably the truth, because not only is the original statement a lie, the ‘truth’ is also a truth. I am not sure how it relates to mathematics, but dammit, I finally figured out that sentence all by myself.[/quote]

Intersting.
Yes the statement must be a lie & could not be made by someone who always tells lies or never tells lies but could be made by someone who sometimes tells lies…I think …

The relationship to mathematics was lost when the thread was split.
The original discussion was around the difference between Fact (the truth ?) & Opinion ie people stating things as fact when what they were actually stating was an opinion based on thie interpretation of the evidence available at the time.
I illustrated this (perhaps poorly) by giving the example of a spreadsheet
on which is written:
1+1 =
The answer would seem obvious & a fact …2.
But then add more information …the numbers are rounded automatically to 1 ie 0.45 = 1 & 1.44 = 1 . Then the answer to 1+1 can equal 1, 2 or 3 .
Hence since we are always working with finite knowledge the answers to questions (even 1+1) will always be opinions rather than facts.

You can then see the relevance of the Bertram Russell quote earlier.
People stating thier opinions as facts (ie irrefutable truths) & interpretting contrary opinions as lies or wrongs (eg a lot of religious zealots)leads to a lot of trouble in the world.

28

To prove or disprove “1+1=2” you have to define what “1” and “2” actually are. “2” could be defined as “1+1”, there’s your proof

Theories can be disproved but not proved. The theory of gravity is just a threoy, you can’t prove something will always fall down, you can do it a billion times and it will work, it doesn’t mean the next time it won’t fall up. But it’s fair to say that’s it’s pretty well tested, although Einstein did adjust some of Newtons theories with realitivity, they are close enough for most everyday situations. Can thorems be proved mathetically ? Pythagoras’ Theorem (Some of the hippopotamuses are on the other 2 sides and all that) can probably be proved with trigometry.

If you want a head ache, try quantum physics. You’ve all see the experiment at school with shining a laser through 2 slits to make a pattern. Light is just a load of photons (Wave/Particle duality bollocks) And the photons interact on the other side of the slits to make a pattern. Weird thing is, you get a pattern if you send one photon at a time… as if the photon goes through both slits at once… They didn’t tell you that at school… I get realitivity, but quantum physics just isn’t common sense (how can something be in 2 places at once ?), but it’s the only way to explain what they see happens

29

Nerd alert!!!

(I’m trying to see my new avatar)

30

You call me a nerd and you have an avatar taken from “The Galileo 7” ? (season 1 episode 14) :?

31

Scuba wrote and this is a fact: People stating thier opinions as facts (ie irrefutable truths) & interpretting contrary opinions as lies or wrongs (eg a lot of religious zealots)leads to a lot of trouble in the world.

Scuba, this is your interpretation.

EB

32

[quote=“speakpigeon”]Scuba wrote and this is a fact: People stating thier opinions as facts (ie irrefutable truths) & interpretting contrary opinions as lies or wrongs (eg a lot of religious zealots)leads to a lot of trouble in the world.

Scuba, this is your interpretation.

EB

---------[/quote] This is my opinion which may be right or wrong. Some people would agree that eg religious intollerance is bad others would not , depends on your perspective . In the same way that the need for revolution is rarely believed necessary by the ruling class. We have to choose our own interpretation & hopefully be mindful that that is what it is…an interpretation & not a fact, thus leaving a door open for our views to be modified by new information.

33

:smiley: :smiley: :smiley: :smiley: :smiley: :smiley: :smiley:

Funniest post of the day.

Brian

34

I’m sorry. I wasn’t calling anyone a nerd except me. I was referring to my Mr. Spock Avatar. As in, “Nerd Alert! look at my Mr. Spock Avatar.”
I know I shouldn’t have put it in this thread, but I didn’t want to start a new thread and I never had an avatar on this board before and I didn’t know if they would automatically take my guanxi out or if I had to manually pay for it because at first I didn’t see it then I added a new post just to see if it would show up and then finally I went to the FX store and paid for it and wow like magic it appeared.

Sometimes I say stupid things but I don’t call people names and I’m sorry that Matthewh took offense.

and ps, 1+1=2.

35

Richardm wrote without taking a breath:
I know I shouldn’t have put it in this thread, but I didn’t want to start a new thread and I never had an avatar on this board before and I didn’t know if they would automatically take my guanxi out or if I had to manually pay for it because at first I didn’t see it then I added a new post just to see if it would show up and then finally I went to the FX store and paid for it and wow like magic it appeared.

Long sentence, isn’t it?

EB

36

1 + 1 = 2?

We may not even be able to say exactly what we mean by that…

EB

37

[quote=“Doctor Rob”]The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.
P2. If x is in N, then its “successor” x’ is in N.
P3. There is no x such that x’ = 1.
P4. If x isn’t 1, then there is a y in N such that y’ = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x’ in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a’
(using P1 and P2). If b isn’t 1, then let c’ = b, with c in N
(using P4), and define a + b = (a + c)’.

Then you have to define 2:
Def: 2 = 1’

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1’ = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn’t 0, then let c’ = b, with c in N, and define
a + b = (a + c)’.

You also have to define 1 = 0’, and 2 = 1’. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)’
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)’ = 1’ = 2 Q.E.D.[/quote]
From mathforum.org/library/drmath/view/51551.html

I wrote before that the proofs are out there. I also said something about not speaking mathematician language. I didn’t mean for that to insult anyone, I just meant that one has to be able to work through all this for it to mean anything to them, it (the proof) means nothing to me. It just has me like :? .

Also to Scuba, the reason I posted the letter, again, wasn’t intended to insult anyone. I know it came from my buddy’s point of view, but the letter not only chilled me out (I was actually upset :laughing: ), but it makes the point that if someone doesn’t want to believe something, they don’t have to (no matter how much “evidence and proof” you have).

1+1=2 becuase my mama told me so.

38

I am no mathematician but isnt the weakness of the Peano Postulate (is that a sexually transmitted disease ?) is that P1 1 is in N is an ASSUMPTION, hence it cannot be a proof ?

39

All the postulates are given and don’t require proof. They are assumptions.

The Peano Postulates define the Natural numbers, positive integers. Thus the proof above is only for the case of Natural numbers.

The real numbers can be constructed from the natural numbers, but I still contend that 1+1=2 is by definition and does not require proof.

I like the website listed above though it has many useful informations.

40

You can assume any N you want. So, if you’re proving 1+1=2, it will be natural to assume an N that is relevant.