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Conway wrote:Let's then consider the above analogy. If I place one thing on the table, and then place another like it but only smaller the sum is two. But then if I placed the same thing , and then placed another like it but only larger the sum is still two. However it is clear to all observers that in both cases the sum is not the same "value", albeit the same symbol. What the object "is" does matter. More specifically the space and value of the object. Math does not work for all scenarios, all doctorates in the subject that I have talked to have agreed on that at least.
I don't see how mass, volume, or any other measurement of size makes any difference. We are not measuring objects, merely counting them. Let's say we're dealing with jugs of milk. You place a half-gallon jug on the table. They you place a gallon jug. If you're calculating the amount of milk, they clearly you have a gallon and a half of milk. But there are still only two jugs.
If you were trying to determine space, volume, etc, then yes the size would matter. But if you simply want to know how many jugs of milk you have, then it doesn't.
Likewise, two men are still two men even if one is a thoroughbred racing jockey and the other is Shaquille O'Neal.
Mass, and volume are a measurement of value not of space. For example black wholes. The most accurate way to count is to include all forms of information related to the object the number is representing, unless, as you say all you want to do is count milk. Again mass and value are not measurements of space. And If I am "counting" my milk because I am thirsty, then yes the space, mass, volume, etc., does matter. All information is pertinent while counting.
-- Updated February 16th, 2015, 8:41 pm to add the following --
Mass, and volume are a measurement of value not of space. For example black holes. The most accurate way to count is to include all forms of information related to the object the number is representing, unless, as you say all you want to do is count milk. Again mass and value are not measurements of space. And If I am "counting" my milk because I am thirsty, then yes the space, mass, volume, etc., does matter. All information is pertinent while counting.
-- Updated February 16th, 2015, 8:42 pm to add the following --
Mass, and volume are a measurement of value not of space. For example black holes. The most accurate way to count is to include all forms of information related to the object the number is representing, unless, as you say all you want to do is count milk. Again mass and value are not measurements of space. And If I am "counting" my milk because I am thirsty, then yes the space, mass, volume, etc., does matter. All information is pertinent while counting.
However it is clear to all observers that in both cases the sum is not the same "value", albeit the same symbol.
This depends on how you define value. If you are counting objects on the table then each has a value of 1 and the sum has the value 2. Introducing symbols just confuses matters. One does not need symbols to count. One object and two objects does not involve symbols. one can use different symbols to represent 1 and 2 but the symbol does not change how many this and that are.
What the object "is" does matter.
Whether it matters depends on what you are doing with the objects, but whether they equal two or three or some other sum has nothing to do with what the objects are that are part of the count.
Math does not work for all scenarios, all doctorates in the subject that I have talked to have agreed on that at least.
But it does work for the scenario you presented. If you are looking for smaller or larger then math is necessary for determining weight or size or volume.
Naturally it depends on how you define value. Obviously I am suggesting value as "containing" more information than normally given when a number is considered. The purpose of counting, is relative to what you intended to do with the things you are counting. The counting and the reason for it are not separable. Unless you are counting for counting's sake. Weight, and volume are not spatial measurements. You may have an X mass, weight, or volume, that is in varying lengths of spatial dimensions. So again it does not work for the scenario I presented. Two small apples are not the same thing as two large apples. The symbol representing them is, but the "sum" is not.
Two small apples are not the same thing as two large apples. The symbol representing them is, but the "sum" is not.
The number of apples is not a symbol. We use symbols to represent numbers. The number of apples is the sum. We get more apple from two large apples than we do from two small apples, but that does not mean we get more than two apples from TWO large apples or less than two apples from TWO small apples. You have not shown that one small apple plus another small apple is not two apples or that one small apple plus one large apple is not two apples. It makes no sense to ask whether the apples are large or small if we are trying to determine the sum of one apple plus one apple. The sum is always two. The amount of apple than one apple or two apples yields varies but the number of apples does not. The sum does not refer to the amount of apple but the number of apples.
The counting and the reason for it are not separable.
Right, that is why a recipe might call for two large apples or three small apples. If we use two small apples we may not have enough for the recipe but that is not because somehow one plus one is no longer two. We have to specify the unit of the count. The unit may be ‘large apple’ or ‘small apple’. The arithmetic sum always remains the same as long as we are counting the specified unit, that is, as long as we know what counts as one.
I should think it obvious from my previous post that I am aware of the difference between a sum, a symbol, and a number. This naturally is what I am debating. A recipe that calls for a large or small apple is useless is it not? One man's large apple is another mans' small apple. Now do you see the problem? Further if you had read my post carefully, I never once clamed that the "symbol/sum" changed. Only that that the "inherent value" that is the "real sum" has changed, depending on the space, and value, of the given things being added. You may find it a better use of you time to stop quoting me. I am aware of what I have said. Again the equation always yields a "2", but the 2 that is given is not always the same. You have just agreed with this by issuing the following statement."
"The arithmetic sum always remains the same as long as we are counting the specified unit"
Now, does a number "without a unit" have a specified space and value? The answer is no. Now does a number with a unit have a specified space and value. No, it either has value, or it has space, "depending on the unit", the symbol or number, being a quantity of the value, or space contained, but in no cases in mathematics do we account for both sets of information, at the same time.
Probably the most interesting reply in the whole thread... a suitable modification of the axioms of arithmetic would certainly produce the result 1+1 = 7; for example, we might modify the definition of the set of natural numbers so that 7 is the successor of 1, and 2 takes the place of 7, and see what follows. Actually nothing would follow because we are just changing symbols. Nothing would change fundamentally.
We could arbitrarily define "7", the class of all bundles of cardinality 7, as the class which naturally follows the class of singletons (all classes containing only one member). This would be possible, but it would entail some rather arbitrary and awkward conventions, and would certainly offend against Occam's Razor. Really, mathematically and scientifically,there is no more realistic or economical alternative than to accept that 1+1=2.
But what's this "Orwellian" stuff? Was he a mathematician?
-- Updated Tue Feb 17, 2015 10:59 am to add the following --
Theophane wrote:Yes, with Orwellian doublethink. 1 + 1 = 2, but it also equals 7 if and when it must.
My apologies, in my second post, I forgot to quote Theophane's reply by way of introduction.
Conway, you are confusing the question of how many with how much. We are all aware that how many does not necessarily tell us how much. We do not know how much milk there is in one container plus another container until it is determined how much there is in each. That each is one is not something we have to determine. We have sufficient information to know that the sum is two, and once it has been determined how much is in each we use the same rules of arithmetic to determine how much is contained in both together.
You can call the answer to the question of how much a sum if you like but there really is nothing surprising in the fact that when we ask different questions we get different answers or that the answer to one question does not serve as the answer to another. Whichever question we are asking and whatever answer we get it is always because one plus one equals two. The answer to the question of how much applesauce we get from two apples is always determined by the fact that one ounce or gram plus one ounce or gram equals two ounces or grams.
Alan Masterman:
a suitable modification of the axioms of arithmetic would certainly produce the result 1+1 = 7
When we play by different rules we get different outcomes. This may have a relevant application as in non-Euclidian geometry, but we do not count according to rules. Certainly we can devise rules for counting and it is possible that some arbitrary set of rules may be of some use, but this does not mean that one plus one equals two only because of some arbitrary set of rules.
Alan Masterman wrote:
But what's this "Orwellian" stuff? Was he a mathematician?
Orwell postulated a state so powerful and omnipresent that they very laws of mathematics were made to kowtow to its will -- sort of. If the state decided to make 2+2=5, it was powerful and influential enough that it could make people believe 2+2=5 and get rid of everyone who disagreed. That, of course, did not mean 2+2 was actually equal to 5, which was part of Orwell's point. The state he postulated feared only the human mind, so it devoted herculean efforts to render humanity incapable of thought or language.
is this significant in this discussion? Only as a reminder of the question of why someone would want to convince anyone that 1+1 adds up to anything different than two. If you could convince someone you could change the fundamental principles of mathematics, you can convince them of anything.
MHopcroft1963 I have at no time said that 1 + 1 = 2, is incorrect. Therefore I am not using "Orwellian double think".
Fooloso4 I agree with all things stated in your previous post. Except for what you consider me to be confusing. I already know you believe that I am confused, such as I believe it is you who is confusing what I am saying. Now understand that what I am saying is that Value, and Space can both be determined in one equation at the same time, for all numbers in any given equation. Thus repeating the "same rules of arithmetic" is not necessary. That is why 1 + 1 = 2, can be "REFUTED", which is different from saying it can be proven wrong. The equation can be refuted because it is flawed, not because it is wrong. It is flawed because it does not declare the value and space together of the given things it is intended to represent.
Conway, precisely so. We remember math books where 10 identical apples are depicted in association with number 10. What if apples are totally different in size, from different trees, ripeness, color, freshness and one of them is poisoned? Then we divide it between 10 kids, and one misfortunate will end up dead, so the answer to a problem would be 1 apple per kid minus 1 kid ( unless responsibility in math does not go farther than just "give, no one forces them to eat apples, even though this is what apples are for). This problem has to have a team of chemists or nutritionists to check safety of the apples. Do we ever have this done before we eat something what looks like apple? No. So yes, answer is 1 apple per kid minus 1 kid, even, to be precise, apples are gone after they are eaten, so answer is 9 live kids, who are happy they had a snack, and very sad about loss of their friend. On of the kids vomits on hearing bad news. So now we have 8 kids who snacked, and one who lost his snack. Time keeps changing the answer. What increment of time the problem's answer is good for? Real life is not happening by math's laws. Only some of it, we should not forget that only some of it, just like short-lived trend in lottery, happens in accordance with math. Math is based on generalisation, detachment from reality, it is like Braille for the blind so they to can enter into the wold of information, have reacher life, more knowledge, but it does not return their ability to see. Braille is as massive and complex as language, but it exists outside of vision. People are handicapped in a face of Universe. They need math to know more, to figure certain set of problems, it's just math does not replace the missing organ, missing ability to see world for what it is. Among blind people there are scientists who, perhaps, are working on solving problem of blindness. The same way in math there must be scientists who, using math, work on actually trying to give people what they do not have, the extra ability to perceive world more efficiently. I consider this to be most honorable path in comparison to simply using math to do what it does.
-- Updated Sun Apr 19, 2015 10:06 am to add the following --
Conway, precisely so. We remember math books where 10 identical apples are depicted in association with number 10. What if apples are totally different in size, from different trees, ripeness, color, freshness and one of them is poisoned? Then we divide it between 10 kids, and one misfortunate will end up dead, so the answer to a problem would be 1 apple per kid minus 1 kid (unless responsibility in math does not go farther than just "give, no one forces them to eat apples, even though this is what apples are for). This problem has to have a team of chemists or nutritionists to check safety of the apples. Do we ever have this done before we eat something what looks like apple? No. So yes, answer is 1 apple per kid minus 1 kid, even, to be precise, apples are gone after they are eaten, so answer is 9 live kids, who are happy they had a snack, and very sad about loss of their friend. One of the kids vomits on hearing bad news. So now we have 8 kids who snacked, and one who lost his snack. Time keeps changing the answer. What increment of time the problem's answer is good for? Real life is not happening by math's laws. Only some of it, we should not forget that only some of life, just like short-lived trend in lottery, happens in accordance with math.
Math is based on generalisation, detachment from reality, it is like Braille for the blind so they to can enter into the wold of information, have reacher life, more knowledge, but it does not return their ability to see. Braille is as massive and complex as language, but it exists outside of vision. People are handicapped in a face of Universe. They need math to know more, to figure certain set of problems, it's just math does not replace the missing organ, missing ability to see world for what it is. Among blind people there are scientists who, perhaps, are working on solving problem of blindness. The same way in math there must be scientists who, using math, work on actually trying to give people what they do not have, the extra ability to perceive world more efficiently. I consider this to be most honorable path in comparison to simply using math to do what it does.
I appreciated your reply very much, thank you. If you are interested I posted one other topic that potentially offers a solution to this, and other axiomatic/philosophical/real, problems with mathematics.
Say, ether are 3 apples on a table, green, yellow and red. I ask person in a room "Give me red apple". How can it be expressed in math? I don't know, maybe Einstein would figure out. Tis simple sentence includes so much information: 1. " Give" is a verb. Verb is a key word for a set of physical actions resulting from brain power. Brain has to be of certain size and ability to respond to verb "give". Brain power in turn is interacting precisely with muscles in human body, calculating changeable spatial location of the body, it's attachment to the bones of particular size and strength, distance to the object ( apple), spatial location of the apple, it's shape, color, graspability, weight, distance to a person who requested apple, ability to hold apple in one's hand for needed period of time while traveling to the person, location of person's arms, hands ( organs for taking objects), anticipation of appreciative words. 2. "Me" means another person has to have program of people and sound recognition installed in his brain, so he's able to identify the producer of the sound "Me" to be a person (same as sound "give") who wants you to serve him. 3. "Red" is key word for perception of a wave length of light. In this case, "red apple", not red table cloth or red bowl in which apple is placed. Delegating the task of color recognition to you means person knows or assumes you can see, and you are not sleeping at the moment or are in a coma. He also knows or assumes that you can hear, that you do not mind to obey, to please, able to move, are of a certain height that allows you to see apple on a table. He, probably, also knows that you are not of a higher rank, not president or a general, or his boss. He understands subordination, or assumes so.
It takes that much programming to even utter this short sentence. It takes vast amount of information to act on this simple request. No one said yet that by living we are constantly doing high math. We are doing more, actually. More than human math possible, but less than our true potential entails.
Ramin22 wrote:Hi. Some claim that '1+1=2' is true. It seems to me that '1+1 = 2' is like a short movie in our head. 2 dots get toghether or something like that. Now if you say this is wrong in general and point out that sometimes '1 + 1 = 3', by pointing out that if you put a man and a woman together, after 9 months there is 3 of them. Then supporters of '1 + 1 = 2', will claim that it was a wrong application of the theory. So they have a winning strategy. There is no way they can lose. They have a movie script in mind, if you make a movie that doesn't end the way their script does, they say it was not based on that script. Which is true. But then saying that their script is true doesn't make sense. Or does it?
Ramin
"Communism is awesome" rings "true" to a communist because he's a communist. 1+1=2 rings "true" to an human because the human is biologically made to be able to calculate.
But 1+1=2 has to be applied to a context in order to be called "true". If you take one bottle plus one bottle you indeed have two bottles, therefore the rule cannot be contradicted, at least in this context. However in the quantum world for example, I doubt such a statement will have any meaning. If you get outside of communism, then communism is not "awesome" anymore, it is simply undefined. Something being "true" is just a feeling you have about it. You think 1+1=2 and your body automatically tells you "Yeah, that's true". You tell that to a tree, the tree won't feel it is true.
Furthermore, there is no absolute context to apply "1+1=2". The bottles are imperfect objects. Nothing is individual enough to be called 1 without assuming something. As said before if you take two piles of clays and put them together, it's still one pile of clay. A bigger one of course, the quantities have been merged but were too imperfect to be called 1. Therefore individualising objects is a flaw, and it is what makes you say "1+1=2 is true".
It's true that in reality not everything is the same. So that what we call "things" in reality can appear in varying quantities.
From there, any system devised can be efficacious, eg. binary, decimal, hexadecimal (or approaches that have not occurred to us yet), just as long as it considers the relativities.
1 + 1 = 2 is a statement of relativity. Maths is all about relativity - the comparison between entities.
The greatness of a nation and its moral progress can be judged by the way its animals are treated—Gandhi.
