What Are Numbers? Philosophy of Mathematics (Elucidations)
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- เผยแพร่เมื่อ 27 ส.ค. 2021
- What is mathematics about and how do we acquire mathematical knowledge? Mathematics seems to be about numbers, but what exactly are numbers? Are numbers and other mathematical objects something discovered or invented? Daniel Sutherland discusses some of these issues in the philosophy of mathematics, explaining some of the basic difficulties involved in trying to say what numbers are and how we come to know them. Are numbers mere mental constructs, something invented by us? If so, then why does mathematics work so well and seem to describe the world? On the other hand, if numbers are "out there" in some sense waiting to be discovered, then what is their status and how do we get knowledge of them? After all, you can't see or touch numbers and other mathematical objects. And unlike ordinary empirical truths, mathematical truths seem to have a quite different and special status: they are a priori, necessary, eternal, universal, and absolutely certain. (My Description)
This is a version of an upload from the other channel. It comes from an episode of Elucidations, a philosophy podcast from the University of Chicago. For more information about the podcast and more episodes, go to: lucian.uchicago.edu/blogs/elu...
Another good introduction to the philosophy of mathematics: • Intro to the Philosoph...
A more advanced talk on philosophy of mathematics: • Philosophy of Mathemat...
#Philosophy #Epistemology #Mathematics
I think one major difference between numbers as abstract objects versus other universals like "blue" or "sweetness", is that "number" has formal, self-consistent definitions. There's no cultural or physiological variation in the way numbers are perceived or used in mathematics. I.e. "numbers are sets recursively containing the empty set (von neumann ordinals) or numbers are successors to nothing. These definitions seem a lot more useful and satisfying than calling numbers an example of another kind of object ("abstract" objects) that we cannot consistently define.
It seems like a lot of the problem comes from using adjectives as if they were nouns.
"Three" or "blue" are adjectives. They are descriptions of things; they are not things themselves.
I took a graduate seminar on Kant with Sutherland years ago. He's excellent. He criticized my mediocre writings in the class very honestly and helpfully, in a way that I've always appreciated.
He also didn't teach you a single useful thing.
@@schmetterling4477What is important?
@@chadbrockman4791 Science, engineering, medicine, farming, art.
@@schmetterling4477He taught him how to think. Philosophy permeates all other disciplines like science, medicine, art, engineering etc.
@@schmetterling4477In which way are science and art useful that does not apply to philosophy?
Just remembered, you should re-upload Ray Brassier's Sellars talk (Nominalism, Naturalism & Materialism)
what will be learnt from that talk?
Extremely interesting. Have to say, I never before realized (40 years as a philosopher) just how profoundly formative mathematics has been on Philosophy. Thank you so much for this.
Except that it hasn't. Philosophers are stull stuck in Plato's rear and that will never change.
@@schmetterling4477That's an immensely infantile understanding of philosophy devoid of any substance. Philosophy and methematics are closely interconnected. Mathematics is based on axioms those axioms are based on philosophy.
@@TravelingPhilosopher They once were. Then Plato showed up and broke the connection between philosophy and reality. ;-)
@@schmetterling4477 Right. The totality of philosophy, the entire history, the vast and diversely rich tradition is refuted by that one assertion. Pretty extraordinary.
@@TravelingPhilosopher It's refuted by the bullshit philosophers have been putting out over the years. :-)
A thing that blows my mind is that we not only in practice, but also in theory can only use approximately 0 percent of all numbers out there. Most numbers are a number followed by a so called point and an infinite number of digits seemingly put out there at random. Imagine you change one digit in Pi. There will be an infinite amount of numbers between this number and Pi. They are called indescribable transcendentals and they are infinitely more numerous than the numbers we can communicate with math.
what you are reffering to is that the set of real numbers is uncountably infinite. But I would not agree with you that we can only use a fraction of all numbers. for example when I refer to a specific function f(x) that is defined over the real numbers, I am in fact using all the real numbers, when making statements about this function
This was blissful to listen to, thank you for sharing your thoughts and knowledge with all of us and myself in particular.
The comments section on every single video about philosophy is a sobering reminder of how much we overestimate our own abilities.
Well Ofcourse ... For most knowledges we need to work with the senses but this doesn't mean that the concept created in our mind isn't abstract.
And with respect to numbers I know some East philosophers who worked a lot in answering all the questions about abstract objects and proving there are thought entitys In the world.
It's time to pay more attention to them and see what they say.
For those without English knowledge is a word that includes the plural so there is no need to say knowledges
Differences ...
Maybe this is the start of brain ...
Yes, treating adjectives as if they were nouns is confusing.
All language is nouns. Adjectives are nouns as well, being funny is the noun of that state. "run" is the noun of that action. "running" is the noun of that action considering the time it's happening.
Quantity is recursive boundary conditions. Math is a subset of logic that deals exclusively with relationships of quantity. Logic is relationships that always replicate. Logic is a subset of science. Science is rigor.
Numbers are individual instances of quantification.
Cool
I would say that science is a subset of mathematics as it is only applied math. Mathematics to me is more like a language and it can be used to speak about things that are not necessarily always applied.
@@catloowitlatkla6116 If the point of science is rigor, numbers are only one way to be rigorous. Of course you can define these things in any way you like, and most people do, but my contention here is that this particular understanding is based on the most meaningful attributes, and thus divisions, of those fields - it does the best work.
Math is definitely a language, useless if it does not describe reality, but the reason it's a subset of logic is that logic does the same thing, describe relationships of things in reality that Always Replicate, just like math, but broader.
@@havenbastion I think DoS does a good job of providing a map of mathematics. Unless you are referring to logic in general, formal logic is a branch of mathematics. th-cam.com/video/OmJ-4B-mS-Y/w-d-xo.html
Edit: at very least I would say the set of mathematics is the same as logic - logic being part of the basis of mathematics, but I don't find it to be a subset.
In the video it is expressly said that mathematics can not be derived from logic alone (21:45) and therefore mathematics is not a subset of logic.
Platonic forms....mathematical objects exist as a universal proposition, P. P exists in mind, but P exists external to human mind as mind independent objects. P exists in the ultimate mind, hence God.
See, there is the bullshit, already.
Interestingly enough those that don't like the words dreamy or vague, prefer to use the Latin equivalent which is abstract, which effectively means distanced, perhaps because dreamy or vague are too realistic and truthful, and Latin words help swagger if I have something in mind that they would dreamy I call it abstract, and hope God that my audience have no Latin or is simply composed of the lower classes that have no Latin, and for them the clue they lie in the related word tractor - ab meaning away or from thus abstract drawn away from, but really will serve as well.
There is nothing dreamy or distant about the absence of large numbers on your bank account. ;-)
Fingers...first experience of quantity
What is a number? A Mathematical Breakthrough
God is the designer of our universe, and He is eternal.
The proof is in the math.
Here is that proof: Infinity = 1/x(delta) + 1.
This equation says a number, any number is a set-in space that change with space.
In physics this equation reads: Gravity is matter changing with space. It combines Relativity or fractured space with Quantum mechanics or spatial expansion.
How dose God fit into this equation?
This equation is God's mathematical name.
God's name in this equation reads: God's Mind Is Man Changed With God.
Breakdown: God's mind is infinite. In math this measure out as the set of infinity
In math (1/x) represents a fraction of a whole. Any child is a fraction of a parent and man according to the Bible is God's child. Therefore, man is a fraction of God
Change in math is represented by the Greek letter (delta) and it denotes a difference of some kind.
Plus (+) in math means: “with” the addition of
There is only one God. In mathematics the number “1” represents a single entity.
Spelled out: God's Mind (Infinity) is (=) Man (1/x) Changed (delta) With (+) God (1).
Scientific Method
Step 1 Observation: Math can deliver unbreakable truths such as 2+2 will always = 4
Step 2 Question: Do math and Divinity share a common truth?
Step 3 Hypothesis: If God exist, He should be found in the house of mathematics.
Step 4 Prediction: God's Mind Is Man Change With God is an equation
Step 5 Test: Any number (Infinity) is (=) a set-in space (1/x) that change (x^2) with (+) space (1))
Note: "X" describes any set, (1) describes any kind of space physical or otherwise
This equation tells us why 2 feet is not the same as 2 inches. Both distances are measured out as 2 units of space but there is a change or difference between both units. They are each sets in a space of distance, but they represent changes in their measurement of distance.
Step 6 Iterate: New look at what makes up reality. Reality consists of 3 domains of space.
a. Fractured space or matter b. spatial expansion a.k.a time and energy c. Complete or unbroken space/information
Step 7 Conclusion: We now know Infinity is real therefore the value in enumeration demand God exists otherwise the domain for enumeration would be incomplete. We know the domain for enumeration is complete because we can count. God must be able to count too all the way to Infinity because His mathematical name tells us what is any number.
Cantor's Mistake
George Cantor known as the father of set theory was wrong. There are no sets of numbers larger than Infinity.
Cantor's mistake was he did not see that "change" is a subset within Infinity.
Cantors one on one correspondence between sets of numbers is wrong. Cantor used only one description of a number from one set to match out or with a number from a different number set.
Example. Cantor said the whole number set was smaller than the integer number set. This is how he made his measurement.
Take the integers 2.1 and match it with the whole number 1. Then match 2.11 with the whole number 2. Then match 2.111 with the whole number 3 and so forth. In this view we would run out of whole numbers when we get to the integer 3.1.
This is Cantor's big mistake!
A correct set correspondence method
Here is a better way to measure these two number sets.
Match 2.1 with say 2. In the next sequence match for 2.1 we could match this integer with 4/2 or 5-3 or the square root of 100 divided by the square root of 25. The point being we can match any description for the number 2 to continue this [integer- whole] number matching sequence forever. In this way we could then match the integer 3.1 with 9/3 or 7- 4. Again, if Cantor had understood that change describes what any number looks like he would have known there can be no numbers larger than Infinity.
Now that we have the knowledge of what is a number. My question is why now? Throughout all of man's conceptual use and beneficial outcomes from using numbers why is it we did not see the anatomy of a number until today? How is it possible that we have been unable to see that numbers do more than describe our physical reality, but they also describe our existence outside our perceived notion of reality. Numbers like truths don't lie.
Yes, we are creatures of the cosmos and whatever makes up the cosmos is in many ways our inheritance. Learning is a part of our cosmos and we do know great discovery comes about over time. There is not always a discovery that changes the world, yet this equation is fundamental to all of existence and it comes from the creator of this existence. So, again why has this knowledge been away from us so long?
Here is my thinking. Mind you my thought in asking then trying to understand this event is not based in math or science but in faith.
We have been blessed, but I also believe we should be concerned for what is coming.
Very highly speculative: Infinite gravity suggest we may be living inside a black hole that is internally expanding. 1/x(delta) may explain why inflation happened. The case maybe that inside a black hole space is cracked and stretched due to the compression and pulling of space by the difference in layer spatial collapsing. Outside space coexisting with points of space already consumed into an infinitesimal boundary create symmetry and this symmetry get to spread evenly as matter. Our universe becomes virtual and expanding. Zero in this context equals the difference in symmetry. A zero field is in this case a field of opposites. One field is collapsing while its opposite is expanding.
This speculation does not rule out God. If it is how our universe happen and is evolving it is best understood as a tool used to do the work needed to fashion existence and life. We should not be afraid of knowing God's working regardless as to how He choose to do those works. Whether it be evolution or any other methodology in His works the truth is we are here to learn and practice those learnings.
If doubt remain then please answer this question. What is a number? Google it if you need help. Infinity says a number is both qualitatively and quantitatively a set-in space that change with space. Isn't this what we do when we count or measure anything at all. Yes, this is exactly what we do when we measure or count anything.
In counting we take a memory or a something we name and put that something into an order of some kind in the space of our mind. We can arrange that something into least to greatest or whatever meets our satisfaction but the fact that we put anything into an ordered sequence is in effect making a change happen.
So, there we have it. A set in the space of our mind changes with the mind. At one moment the set is 1 and at the next moment it is 2 then 3, 4, 5 and so on.
Conclusion why would God give us his love scientifically and mathematically if he did not want us to know HIM? Fighting over whether Creationism or Evolution is the right answer as to why we are here is the wrong picture both have a place with God.
william.mabon@yahoo.com
I am just going to reply to the first part of the Cantor's Mistake portion. There are a few mistakes here and there that mess the whole thing up.
1. Your definitions of Whole numbers, Natural Numbers, Integers, and Rational numbers are wrong compared to the math definition.
2. Cantor proved Whole/Integers/Rational numbers all have the same cardinality (crudely the sets have an equal number of elements) and these are in a 1-to-1 correspondence with the Natural numbers, so these are called countable infinite. Now he did prove that Natural numbers have a lower cardinality than Real Numbers, so maybe that's what you meant?
3. The way he put rational numbers (fractions, or your 2.1, 2.11, 2.111) into a 1-to-1 correspondence with the Natural numbers is completely different than the way you were trying to do it. It is actually in a way that you will hit every single rational number.
Look up definitions of those number systems, Cantor's Diagonalization Proof, Countable and Uncountable infinities and proving rational numbers are countable infinity. Shore up those arguments, you can make your argument tighter, and you'll be good to go
Cantor's mistake is that he considered infinity and zero to be numbers.
In any number system there are axactly 2 with different propety's volumes zero and infinity.Infinity is closed(limited by numbers) but zero is open(unlimited) and numbers lies between them as faces.
1.For example, take binary n-digit number system, it can be represented
by n-vertex simplex. Take triangle.
000 (zero) is externity of triangle
111 (infinity) is internity of triangle
001,010,100 (digital root=1) is vertexes of triangle
110,101,011 (digital root=2) is edges of triangle
2,Take octal n-digit number system.It can be represented by n-vertex simplex behind vertexes of wich lies binary 3-digit number system.
In this case volumes (zero and infinity) of binary 3-digit number system
became faces.Volumes became faces if only you increment dimension.
So i wont to say that volume can became face and face can became volume.But dimension of master (in any number system only 1 zero and 1 infinity other zeroes and infinities from volumes became common numbers) zero+infinity does not changes. If we give context to infinity and zero, for example, space and time (assume space is infinity and time is zero ) then numbers (many,many,many faces) became present. In that present we inceasently going along this faces ( slicing from 1 to 2 from 2 to 3
and so on inceasently around infinity, infinity below each number and zero above each number). In that question you are absolutely right.
I like how you prove God can be anything at all, probably even more than any thing; any thing you want or imagine, or can't imagine. It is inexhaustable.
How about the next question you try to solve is "What is a question?"
@@Ignirium It is some sort of emptiness in mind like zeroes in math or energy in physics or time in philosophy. Nature automaticaly start to fill it with True and False in different proportion. Next question2: what is False and True ?. If you give answer2 then guestion3:
what is FALSE and TRUE of answer2 ?...
@@user-ys3ev5sh3w I've heard your description before :)
I find "God" to be to be quite a fascinating question as an atheist.