Do numbers exist?

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Sat 15 Aug, 2009 12:44 pm
Response from Peter Smith on August 13, 2009 in askphilosopher.org:[RIGHT][/RIGHT]

Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, there must be such things as prime numbers. (3) And if there are such things as prime numbers, then there must be such things as numbers. Hence, from (1) to (3) we can conclude that (4) there are such things as numbers. Hence (5) numbers exist.
Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid.

So that leaves two possibilities. We can challenge the argument at the very end, and try to resist the move from (4) to (5), saying that while it is true that there are such things as numbers, it doesn't follow that numbers actually exist. This response, however, supposes that there is a distinction between there being Xs and Xs existing. But what distinction could this be?
Well, someone could use mis-use "exists" to mean something like e.g. "physically exists": and of course it doesn't follow from there being Xs that Xs physically exist. But it isn't physical existence that is in question when we ask whether numbers exist -- trivially, they aren't the kind of thing you can weigh or stub your toe on! It's granted on all sides that numbers are not physical things.

But once it is clarified that "exists" is not being used in a restrictive way to mean, e.g., physically exists it is very difficult to see what the supposed distinction is supposed to be between there being Xs and Xs existing. Certainly, few modern philosophers believe that there is such a distinction to be drawn. (In a slogan, they think that existence is indeed what is expressed by the so called existential quantifiers, 'there is', 'there are'.) So we'll set aside this challenge.

The other option is to challenge the initial assumption (1). But how can that be done? 11, 13, 17, 19 are the only prime numbers between 10 and 20, and so there are four prime numbers between 10 and 20. That's a simple truth of arithmetic, surely.

But ah, you might say, we need to distinguish: it's certainly true-according-to-arithmetic that there are four prime numbers between 10 and 20, just as it is true-according-to-the-Sherlock-Holmes-stories that Holmes lived in Baker Street. But it isn't really, unqualifiedly, true that there was a man called Holmes living there, and it isn't plain true either that there are four prime numbers between 10 and 20. And from the granted assumption that it's true-according-to-arithmetic that there are four prime numbers between 10 and 20 the most we can reasonably infer is that it should (given arithmetic is a consistent story) be true according to arithmetic that numbers exist. And that doesn't show that numbers really do exist.
This sort of "fictionalist" line that treats arithmetic claims as if claims within a story (the story of arithmetic) has its warm supporters. They will deny that numbers really exist, but at the high price of denying that arithmetical truths are plain true (a reversal, then, of the traditional philosophical ranking of mathematical truths as the most secure truths of all!). If you are not willing to pay that price, and are also not willing to play fast and loose with a supposed distinction between there being numbers and numbers existing, then the simple argument above for the conclusion that numbers exists will look rather compelling.

rhinogrey

Sat 15 Aug, 2009 01:09 pm
@vectorcube,
Was that supposed to be philosophy, or did he just barf on the keyboard?

There's no argument there. He's presupposing the existence of numbers to argue the existence of numbers.

I see 'Peter Smith' has been served well by the public school system.

Theages

Sat 15 Aug, 2009 01:15 pm
@rhinogrey,
rhinogrey;83475 wrote:
He's presupposing the existence of numbers to argue the existence of numbers.

Yes, he's aware of that and devotes nearly half of his exposition to that problem. Great response.

Exebeche

Sat 15 Aug, 2009 05:02 pm
@Theages,
vectorcube;83471 wrote:
Response from Peter Smith on August 13, 2009 in askphilosopher.org:
Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, there must be such things as prime numbers. (3) And if there are such things as prime numbers, then there must be such things as numbers. Hence, from (1) to (3) we can conclude that (4) there are such things as numbers. Hence (5) numbers exist.
Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid.

I guess Smith was a talented mathematician, however not very familiar with logic.
Look at it this way:
(1)There are four archangels in the 40 heavens. (2) But if there are four archangels, there must be such a thing as archangels. (3) And if there are such things as archangels then there must be such things like angels. Hence from (1) to (3) we can conclude that (4) there are such things as angels. Hence (5) angels exist.
Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1) (2) and (3) to (4) is evidently valid.

It's amazing how people who obviously are highly intelligent and intellectual totally fail when it comes to reducing reality to its most simple constituents: Logic.

You could also try to express it in a formal system like:
(1) There are four elements in the quantitiy Q that have the property a. (2) But if there are four elements in quantity Q with a property a, there must be such things as elements with property a. (3) And if there are such things as elements with property a, then there must be such things as elements of quantity Q. Hence, from (1) to (3) we can conclude that (4) there is such a thing as quantitiy Q. Hence (5) quantitiy Q exists.
And so on....
The proof of the existence of quantitiy Q is based on the presupposition that quantitiy Q exists AND (as if that wasn't enough the existence of elements with property a).
Poor reasoning.

Theages

Sat 15 Aug, 2009 05:26 pm
@Exebeche,
Exebeche;83496 wrote:
I guess Smith was a talented mathematician, however not very familiar with logic.
Look at it this way:
(1)There are four archangels in the 40 heavens. (2) But if there are four archangels, there must be such a thing as archangels. (3) And if there are such things as archangels then there must be such things like angels. Hence from (1) to (3) we can conclude that (4) there are such things as angels. Hence (5) angels exist.
Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1) (2) and (3) to (4) is evidently valid.

Your (1) is not prima facie true, whereas Smith's is. Or are you claiming that talk about prime numbers is obviously as dubious as talk about angels? If that is the case, then I also refer you to the second part of the explanation.

Exebeche

Sat 15 Aug, 2009 08:41 pm
@Theages,
Theages;83499 wrote:
Your (1) is not prima facie true, whereas Smith's is. Or are you claiming that talk about prime numbers is obviously as dubious as talk about angels? If that is the case, then I also refer you to the second part of the explanation.

You know that angels are nonsense. As well as i do.
The question is: Does Smith know ?
Try to get me with the example that is based on formal systems.

salima

Sat 15 Aug, 2009 08:42 pm
@vectorcube,
my question would be, if we can doubt the existence of numbers, why would we not doubt the existence of logic? then based on that, how do we use logic to prove there are numbers?

actually i like numbers and logic-i sincerely hope they exist...can we use something else to prove this? they exist as linguistic concepts, and processes we create for various purposes...but are they inherently sitting somewhere in the void before the first human beings evolved? wouldnt they be created on the mental plane as tools with functions that are transferable to the physical plane? might they be considered as processes that evolved over time from experience?

then the question will be asked, does something created in the mind exist? of course it exists in the mind...now does that mean angels and numbers are equally 'real'? maybe the issue is that things can exist that are not real...so far i have not yet heard a satisfactory definition of the word real. to me real is genuine as opposed to counterfeit so i dont think numbers and logic or angels can be debated in this way.

prothero

Sat 15 Aug, 2009 09:43 pm
@vectorcube,
Why can the behavior of the universe best be expressed in terms of mathematical formulas?
Why does theoretical mathematics so often find an application in the real world?
I think there is something more profound about the relationship of mathematics to reality?
It is not clear to me that all mathematical reasoning is based on sensory experience?

Theages

Sat 15 Aug, 2009 09:52 pm
@Exebeche,
Exebeche;83523 wrote:
You know that angels are nonsense. As well as i do.
The question is: Does Smith know ?

Quote:
Try to get me with the example that is based on formal systems.

I have no idea what this means.

jeeprs

Sat 15 Aug, 2009 10:13 pm
@vectorcube,
My take is that everything hinges on the meaning of the word 'exist'. Bear with me while I try and explain.

I don't think the dictionary definition of the term which equates 'to exist' with 'to be' is adequate. I think 'being' has dimensions that are different to 'existence'. I will come back to this.

I would argue that 'exist' is created from two particles: 'ex', to be apart, outside, or separate from; and 'ist', which is to be. So 'exist' is 'to be apart, to be separate from.'

Anything that exists has an identity, in that it is itself and is not something else. Even if two things are apparently identical, they are separate existents, or separate things. To which you can add the observation that all existing things are composed of parts and located in space and time. This latter observation implies that they have a beginning and an end in time also. (You might say 'but what about atoms? Aren't they supposed to be eternal?' To which I would answer, it has not been proven that atoms, in the sense of an indivisible object, actually exist, at least, certainly not in sense of the word as it has been defined here.)

In fact I would argue that 'existence' is the basic attribute of 'a thing'. (Of course there are also imaginary things, but I will leave those aside for now.)

Now let's come to number. By this definition, numbers don't actually exist, or at any rate, they don't exist in the same way as 'material objects'. But they are real. I suppose traditional philosophy would say they are true a priori. That is, their reality can be established by the logic of their relations. Given certain axioms, mathematical consequences necessarily follow. We know this because all minds perceive them the same way, and some minds have a greater or lesser grasp of their nature and reality. So let's say they are apprehended by the mind, rather than being created by it.

Now nobody has succeeded in defining the basis of mathematics in objective reality. As I understand it, this was a major aim of Russell and Whitehead's Principia Mathematic, and one they did not acheive. Then later Kurt Godel came along and showed that no mathematical system could be regarded as wholly self-explanatory:

Quote:
In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory Source

The net effect of all this is that it is still unclear why mathematics is so very effective, and how it relates to 'existence'. It's a mystery, which of course annoys empiricists, because empiricists generally hate mysteries.

My belief is that mathematics, and numbers, relate to the nature of reality itself, which is different to what exists. What exists is only a very superficial aspect of a reality which is inherently 'intelligible' only by virtue of the fact that it possesses a deeply rational nature. So nature itself is 'intelligent' in this sense, or rather 'intelligence is a fundamental characteristic of nature'. Now of course, this was the understanding that many philosophers had, up until David Hume got us to burn all the books of metaphysics, and John Locke declared our mind 'a blank slate'. This is because, in pre-modern philosophy, number was understood by many to be an instance of a real universal. But these kinds of arguments are indeed very hard to understand, and basically modern philosophy began by rejecting all of them.

This is as far as I have gone with this idea, I hope it stimulates futher investigation.

---------- Post added 08-16-2009 at 02:43 PM ----------

I should add, there is a classic essay on this question, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by Eugene Wigner, which ends with:

Quote:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

It is interesting to note that subsequent to publication of this essay (which, I notice, contains 12 instances of the word 'miracle'), a number of critiques have appeared which endeavour to show that the efficiency of mathematics is neither unreasonable nor miraculous. I am not sufficiently qualified to judge whether they have been succesful but I think I at least understand their motivation.

vectorcube

Sun 16 Aug, 2009 12:05 am
@rhinogrey,
rhinogrey;83475 wrote:
Was that supposed to be philosophy, or did he just barf on the keyboard?

There's no argument there. He's presupposing the existence of numbers to argue the existence of numbers.

I see 'Peter Smith' has been served well by the public school system.

Can you be more specific? Which part do you have a problem with?

Exebeche

Sun 16 Aug, 2009 05:04 pm
@jeeprs,
salima;83524 wrote:
my question would be, if we can doubt the existence of numbers, why would we not doubt the existence of logic? then based on that, how do we use logic to prove there are numbers?

The mathematical terms that you normally look at are based on a particular formal system.
We could point at the decimal system as the most important one.
All of these mathematical expressions could also have a different look, for example in binary or octal numeral systems.
The same numbers look completely different in different numeral systems.
Also mathematical operations like adding or subtracting can look so different that you would hardly realize that it's the same operation.
Basically however the content stays the same.
Only the look and feel changes. That's something miraculous about mathematics.
And it's that thing that stays the same what people call logic.
No matter what kind of formal system you use, the same rules appear in all of the systems.
It would be false to say that these rules are what's logic, but there is consense that these rules mirror at least a part of what is considered the realm of logic.
So, mathematics is actually not the most superior discipline. It is still subject to logic.

Exebeche;83523 wrote:

Try to get me with the example that is based on formal systems.

Theages;83529 wrote:

I have no idea what this means.

Sorry if i did not make myself clear enough.
But by twisting the words in my mouth you were trying to get me with my back to the wall.
That's why i said, try to get me with the formal logics.
In fact i haven't even made a statement about wether or not numbers exist so far.
What i can say however is, that the deduction of Smith is poor reasoning.

Actually i do consider numbers to have an abstract existence.
However Smith's 'proof' is worthless.

jeeprs;83531 wrote:

My belief is that mathematics, and numbers, relate to the nature of reality itself, which is different to what exists.

I see some wisdom in these words.
My point of view is that reality is constituted by more than matter and energy. I am not talking about anything esoteric here, i refer to Norbert Wiener's sentence that information is the third quantitiy in our universe, next to matter and energy.
I am talking about non-energetic relations. Relations that are not energetic or material to be more precise.
It's actually not even difficult to understand:
When we use a word like 'family', or even better 'herd', e.g. talking about a pack of baboons, the word herd describes a phenomenon that is neither material nor energetic.
It's the relations between the baboons that makes a number of animals a herd.
Matter and energy may be what the world is made of, but the way things relate to each other is what gives meaning to it all.
'Panta rei' - Heraklits words express that everything is subject to dynamics. Everything is in flow, the universe is a continuous process (as oppose to a fixed environment).
The flowing, the dynamics are what causes relations to appear.
Any description of the universe that takes in account only energy and matter will remain incomplete (oh, that reminds of our discussion about quantum physics...), it's the PROCESS that causes relations to appear.
If the universe was static (like a picture), there would be no way for relations to appear. But since the universe is in movement, things get in contact and thus relate.
This might even be the logical basis of what (physical) information is.

Back to numbers:
The world of matter and energy is subject of change (dynamics). However matter and energy don't behave arbitraryly.
Everything is subject to certain rules.
Now, anything the human mind perceives is transcripted into particular patterns - neuronal patterns in the first place, that get transformed by humans into formal systems:
Language is a formal system.
Another one is mathematics.
Mathematics is a transcription of the way things relate to each other on a logical basis. Maybe the most abstract one.
So mathematics is a way of describing the non-energetic and non-material relations.
(It can be used to describe energetic and material relations though.)
Numbers are just elements of the description.
Numbers are a transcription and descripton of something that has its equivalent in reality.
They are reflections.

Theages

Sun 16 Aug, 2009 06:20 pm
@Exebeche,
Exebeche;83641 wrote:
No matter what kind of formal system you use, the same rules appear in all of the systems.

Quote:

Sorry if i did not make myself clear enough.
But by twisting the words in my mouth you were trying to get me with my back to the wall.
That's why i said, try to get me with the formal logics.
You didn't clarify yourself, you just repeated yourself. I still have no idea what you mean

Nevertheless, I will explain Smith's point. Take a statement like "There are angels in heaven." You and I both take this to be false. In other words, we both assent to the statement "It is not the case that there are angels in heaven." There is no problem here. I (and you, I take it) will confidently and without reservation assert that it is not the case that there are angels in heaven.

Now, take "There are four prime numbers between 10 and 20." Smith assumes that it is true. In fact, any non-philosopher with any knowledge of math at all will assent to this. He takes it as an established fact. As he and you are both aware, he assumes existence to prove existence. You say this is faulty reasoning. Does this mean that he cannot rely on (1)? Is (1) false? Can you say with any confidence "It is not the case that there are four prime numbers between 10 and 20"? I certainly can't. Most people would think that someone would have to be ignorant or insane to assert that.

The problem is that a wide variety of mathematical statements seem to be obviously true, and this includes existential statements. You can deny the existential statements, but at the cost of severe dissonance with what is presumed to be obviously true.

jeeprs

Sun 16 Aug, 2009 07:04 pm
@vectorcube,
you are not addressing the meaning of the word 'to exist' and whether it applies to numbers, which is the original question.

Also I respectfully suggest that neither you nor I know whether or not there is Heaven, nor angels in it. But there might be. So using this for an example merely illustrates something about your beliefs. There are plenty of other truly fictional examples (e.g. square root of two, the horns of a rabbit, etc.)

---------- Post added 08-17-2009 at 11:22 AM ----------

Interesting article here, on the difference between reality and existence. It also argues that atoms don't exist.

prothero

Sun 16 Aug, 2009 07:33 pm
@vectorcube,
I believe that Russell and Whiteheads program in the principia mathematica was to try to show that mathematics were reducible to logic.
I believe they failed in this effort.
Subsequently I believe it has been shown that no formal system is possible without one or more axioms which are just postualted and accepted.

In any event what really requires explaining is why mathematics seems to explain the workings of nature.

Also why human reason without recourse to sensory input can construct systems of mathematics (a priori) which correspond to the workings of external reality.

Arguments about what is "real" and what "exists" are always circular in nature.
Efforts to reduce philosophical arguments to formal or symbolic logic are in some ways always unsatisfactory, incomplete and flawed.

Theages

Sun 16 Aug, 2009 07:38 pm
@vectorcube,
jeeprs;83647 wrote:
you are not addressing the meaning of the word 'to exist' and whether it applies to numbers, which is the original question.

Also I respectfully suggest that neither you nor I know whether or not there is Heaven, nor angels in it. But there might be. So using this for an example merely illustrates something about your beliefs. There are plenty of other truly fictional examples (e.g. square root of two, the horns of a rabbit, etc.)

I respectfully suggest that it is literally impossible to prove a negative existential claim, so your "there might be" is completely vacuous. I am completely confident that it is not the case that there are angels in heaven.

Also, the square root of two is no more fictional than any number. And what exactly is the difference between being fictional and being "truly" fictional?

jeeprs

Sun 16 Aug, 2009 07:50 pm
@prothero,
prothero;83659 wrote:
Arguments about what is "real" and what "exists" are always circular in nature.

Beg to differ actually. I think the idea that 'reality' and 'existence' might be different has not been discussed much.

I believe it can be used to demonstrate that the idea that 'reality consists of objects' is mistaken. Yet this is fundamental to the realist outlook. So the implications are actually profound.

Let's see how it develops.

salima

Sun 16 Aug, 2009 08:04 pm
@Exebeche,
Exebeche;83641 wrote:
The mathematical terms that you normally look at are based on a particular formal system.
We could point at the decimal system as the most important one.
All of these mathematical expressions could also have a different look, for example in binary or octal numeral systems.
The same numbers look completely different in different numeral systems.
Also mathematical operations like adding or subtracting can look so different that you would hardly realize that it's the same operation.
Basically however the content stays the same.
Only the look and feel changes. That's something miraculous about mathematics.
And it's that thing that stays the same what people call logic.
No matter what kind of formal system you use, the same rules appear in all of the systems.
It would be false to say that these rules are what's logic, but there is consense that these rules mirror at least a part of what is considered the realm of logic.
So, mathematics is actually not the most superior discipline. It is still subject to logic.

Back to numbers:
The world of matter and energy is subject of change (dynamics). However matter and energy don't behave arbitraryly.
Everything is subject to certain rules.
Now, anything the human mind perceives is transcripted into particular patterns - neuronal patterns in the first place, that get transformed by humans into formal systems:
Language is a formal system.
Another one is mathematics.

Mathematics is a transcription of the way things relate to each other on a logical basis. Maybe the most abstract one.
So mathematics is a way of describing the non-energetic and non-material relations.
(It can be used to describe energetic and material relations though.)
Numbers are just elements of the description.
Numbers are a transcription and descripton of something that has its equivalent in reality.
They are reflections.

i think your explanation of what numbers and mathematics are is great. i always felt that the human mind designed mathematics as an expression of something, yes that would be relationships as you have said. thank you!

then what is logic? is it another formal system? other formal systems are subject to logic? has the human mind designed logic or is it inherent in the working of the mind? isnt it in fact a faculty of the mind?

---------- Post added 08-17-2009 at 07:45 AM ----------

prothero;83659 wrote:

Also why human reason without recourse to sensory input can construct systems of mathematics (a priori) which correspond to the workings of external reality.

can you explain what you mean here? i must be not understanding something rightly. is it possible to use the power of reason without taking into account or being influenced by that which has been observed through the senses?

are you referring to such things as how did einstein come up with the theory of relativity? i would assume it was through the power of reason since he couldnt prove it mathematically. and though he cant have had any sensory data on these workings...hmmm. :perplexed:

prothero

Sun 16 Aug, 2009 08:46 pm
@salima,
salima;83668 wrote:

can you explain what you mean here? i must be not understanding something rightly. is it possible to use the power of reason without taking into account or being influenced by that which has been observed through the senses?

are you referring to such things as how did einstein come up with the theory of relativity? i would assume it was through the power of reason since he couldnt prove it mathematically. and though he cant have had any sensory data on these workings...hmmm. :perplexed:

Stanford Encyclopedia of Philosophy "The dispute between rationalism and empiricism concerns the extent to which we are dependent upon sense experience in our effort to gain knowledge. Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience. Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge.

Rationalists generally develop their view in two ways. First, they argue that there are cases where the content of our concepts or knowledge outstrips the information that sense experience can provide. Second, they constuct accounts of how reason in some form or other provides that additional information about the world. Empiricists present complementary lines of thought. First, they develop accounts of how experience provides the information that rationalists cite, insofar as we have it in the first place. (Empiricists will at times opt for skepticism as an alternative to rationalism: if experience cannot provide the concepts or knowledge the rationalists cite, then we don't have them.) Second, empiricists attack the rationalists' accounts of how reason is a source of concepts or knowledge."end of quote

I am a rationalist and feel the ability to reason exists a priori independent of sensory data. The mind IMHO is not a complete blank dependent on sensory input for all knowledge. The mind imposes structure of the eternal world (space, time, causality) and reasons independent of it. I think logic and math are a priori forms of knowledge and results of reason.

Einstein was a theoretical physicist. Most of his greatest contributions were the result of imaginary thought experiments not the result of experiments or the analysis of empirical data. His greatest papers were published while he was working as a patent clerk. He could not find a teaching position in a major university.

"Imagination is more important than knowledge" an Einstein quote. The greatest scientific advances often seem to be the result of creative insight or imaginary leaps more than the result of empirical data collection or analysis. Making science in some ways similar to art.

---------- Post added 08-16-2009 at 08:55 PM ----------

jeeprs;83664 wrote:
Beg to differ actually. I think the idea that 'reality' and 'existence' might be different has not been discussed much.

I believe it can be used to demonstrate that the idea that 'reality consists of objects' is mistaken. Yet this is fundamental to the realist outlook. So the implications are actually profound.

Let's see how it develops.

Alright I will bite. Maybe should start a different post.

Just as a starting point, I think it is wrong to defiine "reality" in terms of material objects or to limit it to what we can perceive with our senses or our instruments.

Thus total reality is much larger than perceived reality or even scientific reality. In fact much of reality may lay outside the bounds of human knowledge or perception entirely.

Things may thus "exist" of which we have no knowledge or perception
and even perhaps no possible knowledge or perception. However what does it mean to say "something exist of which we have no knowledge or perception? It would not exist for us would it? Maybe only in imagination.
Do thoughts exist. Do unicorns exist because I can read, draw and imagine them? Are they part of reality?

Spinoza said God had infinite attributes of which only two (mind and matter) were available to human thought and perception. Are those attributes real? Do they exist?

We have already offended all logical postivists, linguistic analysts and analytic philosophers. For them even the question is meaningless?

I am checking out for a while, have to do some actual work, bummer:o

jeeprs

Mon 17 Aug, 2009 03:53 am
@Theages,
Theages;83661 wrote:
I respectfully suggest that it is literally impossible to prove a negative existential claim, so your "there might be" is completely vacuous. I am completely confident that it is not the case that there are angels in heaven.

Also, the square root of two is no more fictional than any number. And what exactly is the difference between being fictional and being "truly" fictional?

Fair point, I stand corrected.

I would be interested to know if you have any criticism of my suggestion that there is a difference between 'existence' and 'reality' and whether you think this is meaningful.

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