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Dr. Tsvi
Saks' lecture on Mathematical Infinity and Creation - Unedited
The title
that I saw was "Bringing Moshiach Into the World of Science." It
sounded to me like it was l'mala l'mata, from above to below. Bringing
G-dliness down or spreading out G-dliness and Moshiach into the world of
science. I want to do that. I also want to try to do the reverse or the
dual of bringing science into the world of Moshiach and G-dliness. I want
to do this with some very concrete examples. I'm not going to give lots and
lots of different examples. I'm going to pursue a certain line of reasoning
for a while. Stick to certain topic. If you saw the publicity, I was listed
as doing research in artificial intellegence which is true but that is a
second career that I have. The first career that I had was doing research
in pure mathematics. Mathematical infinity. That is the science that I'm
going to talk about today.
I want to
give some concrete examples, as concrete as possible. This is a very
difficult subject to talk about with the public. On the other hand, I'm
going to give it a try. So I'm going to actually discuss some reasonably
technical mathematics. Hopefully it will be successful to everyone in the
audience. If it isn't, just relax because I will not do it forever. I'll
come back and relate it to the Chassidus.
As a
brief introduction to the subject of mathematical infinity, let me pose a
very simple question. You have a party. You're making a melave malka.
Everybody has to wash at melave malka so you make sandwiches and you really
don't know how many people are coming and you don't really know how many
sandwiches you are making. Everybody takes one. That's the rule and
everybody follows it. At the end of the melave malka after you bench there
are no sandwiches left. Everybody took exactly one sandwich. Now what
inference can we make. What information do we have now that we didn't have
before? The fact that everyone took exactly one sandwich and every sandwich
was taken by exactly one person. Do we know how many people came to the
party? We certainly do not. Seventeen, forty-three? It doesn't make any
difference. We don't know anything about that yet. There is something that
we do know. That is we know that the number of people that came to the
melave malka and the number of sandwiches are the same. Because we have a
one to one correspondence.
I was a
college professor for many years. Now I work commercially. I got a real
job. So I forgot to erase the black board when I started because I don't do
this any more.
I'm just
going to draw a very simple diagram of one to one correspondence. If this
is the people and this is the sandwiches. That means the first person took
the first sandwich, whoever they were. There is a relationship here. The
same for three. I'm going to talk about infinity so I'm going to draw lots
of dots. But here the dot, dot, dot is just some arbitrary end that we
don't know yet. This is a one to one correspondence between people and
sandwiches, drawn in some abstract form. I'm just trying to demonstrate the
idea of one to one correspondence. That the number of people and sandwiches
are the same because we can establish a one to one correspondence between
the two co-actions. This concept of one to one correspondence was used by a
mathematician known as George Kantor to define the definition of number.
For finite numbers is not all that interesting. For infinite numbers there
are many, many surprises. So that is what I want to talk about. The way we
generate and the most important sequence which is the natural numbers; the
positive integers 1, 2, 3, etc. is--You start out with one and then there
is a principle or a rule that allows us to generate new numbers from the
old numbers is you can always add one to a number. You start out with one
and add one to it you get two. You have two, you add one to it you get
three. You have N. You add one to it you get N + 1. And the dot, dot, dot
that I just wrote is very important because it means ad infinitum. It means
that
there are actually infinitely many numbers in the sequence 1, 2, 3 dot dot
dot ad infinitum. On the other hand the way that sequence is usually looked
out is its infinity as a potential. We haven't said yet that there really
exists an actual infinite set that exists in completion. The sequence is
potentially infinite. It is infinite because there is no largest number.
However far out in the sequence we can always go out one more. So it is
infinite but it is considered infinite only in the potential. There may not
actually be an actually set or collection of entities which do exist and
co-exist simultaneously as infinity. In fact that was the prevailing view
for a long time, that there is actually you existing infinity. I'll get to
that in a few minutes. What I want to do though is to utilize the view of
one to one correspondence to show you a very strange example involving
mathematical infinity. As I already said, mathematical infinity involves a
lot of surprises. But there are some very strange things that happen when
you talk about infinite numbers. O.K. I want to change the numbers here and
the sequence and relate each number to its square. So one relates to one.
Two relates to four. Three relates to nine. N relates to N square. What do
we have here? What we have is a one to one correspondence between the
sequence "1, 2, 3 ..." and the sequence "1, 4, 9, ..."
Now what is wrong with this picture? What's wrong with it is that on the
bottom the set is distinctly smaller. Two and three are skipped. Five, six,
seven and eight are skipped. Ten, nine, eleven, twelve, thirteen, fourteen,
fifteen are skipped. So we have established a one to one correspondence the
sequence "1, 2, 3, ..." and "1, 4, 9 " which is
distinctly smaller. In other words these two sets should have the same
number of elements even though one is distinctly smaller then the other
one. It turns out that it should be strange to you but on the other hand it
later was used as the basis for discussing mathematical infinity in a
logically consistent way. This example was first discovered by Galileo in
1620 and its interesting that the theory of Kantor are the major
breakthroughs in the science of mathematical theory doesn't come until
almost the end of the 19th century. Over 250 years later which is an
enormous amount of time considering the initial insight was...from the time
that the initial insight was made until the theory actually grew into a
full grown theory. One of the things that Kantor proposed are that infinity
actually exists. And the sequence "1, 2, 3, ..." is not only just
a potential infinity, which is potential infinity in the sense that it has
no largest number but it actually exists as a real infinite collection that
exists in some form. Kantor actually...this discussion in the literature--the
disagreement literature whether or not he was Jewish but he was a very
spiritual person and he had a very, very deep belief in G-d. He believed
that mathematical infinity existed because it was a manifestation of G-d's
greatness in the world. The number he used to categorize this sequence,
"1, 2, 3, ..." is our zero. This caused many people to believe he
was Jewish because he used the hebrew letter as the name of the first
infinite number which he called, "alef-zero." Alef-zero is the
number of elements in that sequence, "one, two, three, dot, dot,
dot." Its an infinite number and it has a lot of interesting
properties which are very different from the finite numbers that we are all
used to. One of these properties is the concept of a predecessor which is
that, just like two, for example, generates three or three generates four
or N generates N + 1, so each number has an immediate successor. Conversely
each number has an immediate predecessor. So three has an immediate
predecessor which is two, which is the number that generated it. Four has
an immediate predecessor which is three, which is the number that generated
it. Alef-zero has no immediate predecessor. The whole sequence generates
it, so to speak. All together it requires the whole infinite sequence to
generate Alef-zero. I'm making that point now because I'm going to come
back to it in a few minutes. Another point that Kantor made is that
mathematical infinity is a limited form of infinity. In fact there are
different levels of mathematical infinity. Alef-zero is only the smallest.
There are much bigger ones and ....its...actually the same is true for the
finite numbers, that there is no largest number. By infinite numbers its
more of a surprise because people think of infinity as something that must
be unbounded. And yet in Kantor's theory, which is generally accepted my
most mathematicians these days, there is no largest infinite number. Every
infinite number has one that is larger then it. So that mathematical
infinity is an infinity in a sense that it is not finite but its not
absolutely infinite like G-d Al-mighty is infinite. In fact, Kantor was
able to prove another limitation which is that the totality of all objects
that exist cannot be given a number. Its too large. The totality of everything
which exists is too large to be given a number and again that is just a
further expression of the limitation of mathematical infinity but it is
still infinite. Actually Kantor called it "trans-finite" perhaps
to avoid the word finite. I'll come back to those points again. I'm just
laying the groundwork here in this brief discussion of mathematical
infinity so that when I relate it, with G-d's help, to Chassidus I'll have
something concrete that I can talk about.
Let's
move over to some Chassidus. The Creation as discussed in Chassidus as most
of the audience knows here, is not that G-d just said, "poof" and
the physical world came into being. He created a panorama. An infinite
panorama of spiritual worlds. There is a concept of infinite descent of worlds
where descent in the sense of that any...at any particular level of a
spiritual world--of a olam, so there is a contraction. There is a
concealment giving rise to a next lowest world in which there is more
concealment of G-dliness in which the world and spiritual beings that
inhabit that world have more sense of self and more self definition and
more individuality. And the culmination of the process, of that infinite
[collective] descent of spiritual worlds is the existence of the physical
world. This is--another term that is used here is the concept of yesh
mayayin--creation of something from nothing.
What does
something from nothing mean? And how is this creative process described?
When G-d Al-mighty created the world he did it by putting a spiritual life
force in every created being which is the source of the existence of that
being and continues to keep it in existence after it was originally created
and continuously recreated it. So everything that exists as a creation has
G- dliness in it which is the source of its existence. So the question is
asked in Chassidus, why is it called "something from nothing?"
The physical world is a transitory existence. A dependent existence should
be called, "a nothing." G-dliness which is the source is a true
existence. That is where we have something. It should be called,
"nothing from something." Why is it called "something from
nothing?" So its called "something from nothing" because the
"something" thinks its a something. Because the something is
unaware of the fact that it has a source, it needs a source, it has
G-dliness within it which created it and keeps it in existence. So the
concept of yesh m'ayin, of something from nothing ex nihlo is that the
something cannot trace its existence back to its source.
My claim
here is that the mathematical model that we have here is a model for this.
Because this number Alef-zero is yesh m'ayin. It cannot trace itself back
to its source. Its source is the entire infinite process which is coming
into it. The sequence, "one two, three, dot, dot, dot" and I can
certainly visualize that from its perspective, from the perspective of this
guy there is no immediate predecessor. It doesn't have a source that it can
lay its hands on and he says, "I'm independent." And be
completely unaware of this infinite process that comes into it. Just like
the physical world and people who, for what ever reason, G-d forbid, don't
believe in Hashem, and are not aware of the concept of the creation see
themselves as being independent because they don't have anyway to relate to
the infinite process which is coming into them at every moment and
continuously recreating them. So we have a model up here of the Creation.
The earlier concept that I mentioned of the infinite descent of
worlds--don't get hung up on the numbers, "one, two, three". Just
think of them as place holders for some entity or each one of them could be
a spiritual world. So one is the highest, in some sense, the succeeding one
is lower, in some sense. And that infinite process, that infinite descent
of spiritual worlds culminates in the existence of the physical which in
the model is the number Alef-zero.
That is a
very simple model for the Creation and its too simple for a number of
reasons. So I'm going to try to expand it. There are many questions from
different ma'amorim that can be asked about this model that would require
us to expand it. I'm just going to do one of them because the mathematics
would get out of hand if I try to do too much of this. But for example, we
know that there are four major worlds. There's Atzilus, the world of
emanation, Breyah-- Creation, Yetzira--formation and Assiyah--the world of
action, the physical world. There are infinite gaps between them. Atzilus
in infinitely higher then Breya. Breya is infinitely higher then Yetzira
and so on. Well, we don't have that here on our model because the descent
or the sequence which culminates in the existence of a new object at the
end of it is only a gap of one in each case. So that is not infinite. So
what I want to do is to expand this model into a more complex one in which
will have in it the notion of infinite gap. In order to do that I need to
talk about the concept of ordinal number. Ordinal number means numbers
which are based on ordering as opposed to cardinal numbers or quantitative
numbers which are pure quaniting. For example, its a difference between
seven and seventh. We know that the Rebbe, Shlita, is the seventh
Lubavitcher Rebbe. Kol shvi'im chavivin. All sevenths are beloved. The
Frierdiker Rebbe made a big deal out of the fact that the Rebbe is the
seventh Rebbe in last Ma'amer, Basi L'Gani. So the difference is there are
several Lubavitcher Rebbes. When we say that, there is not implication at
all of first, second. They are lumped together as equals without any
distinctions being made on any of them. But if we say that the Rebbe is the
seventh then there must be a seventh of the first as the Rebbe Shlita
himself talks about in his first Basi L'Gani ma'amer. The seventh the first
to the Alter Rebbe in this case. So Moshe Rabbainu was the seventh from
Avraham Avinu. That is the difference between ordinal numbers which an
example is the seventh and seven which is just a group collection without
any ordering on them.
For
finite numbers, its basically the same. There is nothing remarkable about
the distinction. For infinite numbers there is something remarkable because
the ordinal numbers are much more subtle then the cardinal numbers in the
following sense: If we add one, quantitatively to Alef-zero, the first
infinite number, then its still Alef-zero. Alef-zero plus one equals
Alef-zero as a quantitative. In fact the Alter Rebbe is going to say that
almost exactly in Tanya Chitas this coming week. Where he says if you add
any finite quantity to an infinite quantity it doesn't make any difference
its kol [hebrew ] sheviv. Its considered like absolutely nothing. This is a
mathematical interpretation of that concept. Ordinary its different though.
Ordinary, we can add one to Alef-zero and get something new. So I'm going
to draw Alef-zero plus one. O.K. now this [area] is different because as
ordinal numbers, as the way that the number in question relates to what
came before, Alef-zero is the end point of this whole extension of this
infinite sequence of numbers. The "one, two, three, dot, dot,
dot" gives rise to the existence of Alef-zero. Alef-zero sits on the
end and contains that until our sequence, in effect, which defines its
particular properties that it has no predecessor. It has not immediate
predecessor. Where as Alef-zero-plus-one has an immediate predecessor. Its
immediate predecessor is Alef-zero which is the limit point, the end point
of the infinite sequence. So as considered ordinarly these two are very
different despite the fact that quantitatively they are identical. That may
seem strange but that is the way it is. Once you start adding one to things
then we can keep going. I really need a blackboard that can extend all the
way out there somewhere but pretend that this is winding around now. Let me
rewrite Alef-zero. Alef-zero-plus-one. Alef-zero- plus-two. Eventually you
get to two[ ] zero You can add one infinitely many times and get to three-[
]-zero. You can add one infinitely many times and get to four [ ] zero.
Eventually you get to something even larger. If you are not following all
the details here, don't worry about it. Eventually you get to something
which is Alef-one. That is what Kantor called it. What is Alef-one?
Alef-one is the first one of the numbers in this sequence which is quantitatively
bigger then all of the others. All of the things that came before are
quantitatively the same but in ordering they are different as ordinal
numbers but Alef-one is the first one that is quantitatively bigger then
all of the rest that came before. This is a better example of a model for
the Creation as explained in Chassidus because...[questioner in background]
How do you know it exists? Lets talk about that after. I can answer it but
I'm in the middle of something. So the point I want to get to is you have
now a much longer infinite sequence and here we have an infinite gap. Here
we have an infinite gap. Here we have an infinite gap. An even larger
infinite gap. So we have a sequence which incorporates the concept of major
stopping points along the way with infinite gaps between the major stopping
points. So for example this could be Atzilus, this could be Breya, this
could be Asirah and this could be the physical world. I don't know the
exact math. At last years conference a statement was made that Yisroel
Aryeh Leib could discuss the whole seder histolshulus, the whole creation,
from Bina down to the physical world in English so anybody could understand
it. So, G-d willing, he should be with us with tehiyas hamasim and he can
do that. I honestly believe that if he could do that then we could get some
mathematicians together and really ...give very precise mathematics to
describe that process as a model. I know that we can't do that. The model
is still vague. But I think its still a pretty good model. It amplifies the
point of an infinite sequence with major stopping points in it and it gaps
between the major stopping points. I should also probably say that the
models that I worked with as a professional mathematician were incomparably
more complicated then this. Really, where you go into [ ] as the first two
letters of the mathematical [ 326 ] alphabet. So the models I thought I
could use to model the concept of yesh mayaim or seder histokulos, the
Creation of spiritual worlds were incredibly more complicated then this and
had a lot of properties which you can look in ma'amorim and see, yeah the
model contains that property. Another concept or a related concept to all
of this is the concept of [335 al ...mispar]. Actually it was in Tanya
yesterday. What does it means that there were worlds [ada mispar] without
number. I think what it means is that [ 338 ]. That infinity of worlds, the
Alter Rebbe said today in Tanya that it is not comparable to the absolute
infinity of Hashem Himself. Its [hebrew ] its like absolutely nothing
compared to Hashem Himself. And yet it is infinite. There are infinite
number of worlds. My belief is that its mathematical [ theory?] is the
right language to describe it. Again I don't have all the answers but I
think that it is the right language and the right context because
mathematical infinity, like the concept of infinite number of worlds
expresses and infinity which is limited and makes no claim of being
comparable to the absolute infinity of Hashem Himself. That was a very brief
introduction to the idea of using mathematically infinite models to
describe concepts in Chassidus. Let me talk about it a little bit
personally because this was a big deal in my life. My p.h.d. thesis and the
subsequent research that I did had literally no relationship to the
physical world. No application. Nothing that any one knew of that the
mathematics would model and I always wondered, like anyone who did that
work would wonder, what application could there possibly be for this
mathematics. When I came to become observant and a Lubavitcher....I should
have mentioned at the beginning of the talk that there were numerous
professors who asked the Rebbe if they could leave their positions and go
and learn in yeshivah. I took a different route. I didn't know at the time
to know that I should ask the Rebbe if I should do it. I just wrote to the
Rebbe that I was doing it. And I did leave my position as a professor in a
university and went to learn for three years, in Hadar Hatorah which is
just across the street. I would really like...really should express
appreciation to the yeshivah because I lived there. I had a little room. I
had my three meals a day and excellent instruction from dedicated teachers
and didn't have to pay anything and its a wonderful thing. Anyway, sitting
in that yeshivah and starting to learn Tanya and learning Chassidus, the
descent of worlds and I was just overcome by the knowledge and the
realization that this was the actual application of the mathematical work
that I had worked on for so many years that had no other application. That
was an amazing experience. To give it a little more context, before I
became observant, I was a product of the 60's. Was a so called spiritual
seeker and several people who I knew from the old days who became observant
before I did, so they told me, "all that stuff that you ever learned
in what ever discipline it was, Chinese or l'havdil whatever it was, its
all in Torah. They all have their kernel of truth. And Torah is the one
that contains them all. I heard it enough times but I believed it. I
expected it to happen. And it did happen. I can tell very precise stories
about things that I learned from other disciplines that were very useful at
the time. And in Torah, it was in a ma'amer. It was in context of a
whole...you know...those who learn Torah and Chassidus. Its part of a whole
system and its with depth. It was much, much more then I ever had before.
That was all that I expected. But that the actual application of the
mathematics I did was in Torah--that was something that I was totally
unprepared for. I could never had imagined and that was a truly awesome and
inspiring experience. Let me move on. So I'm involved in applying
mathematics to Chassidus. I spoke at some D'vorah Torah conferences that were
in Miami. That was very useful. It was inspiring. It gave me a chance to
[418 ] and speak publicly about these ideas that were germinating inside of
me. They are not so easy. It took a long time to the point...I don't know
if anybody understood what I've said so far but it took a long time to get
to the point where I could take the ideas that I had which were very, very
abstract and very difficult and bring them down to a point where I could
actually talk about it more or less in English to a non- mathematical
audience. In any case something happened in tav shin nun ches. The Rebbe
published a ma'amer. Chof-bais shvat tav shin nun ches. Its starts out with
Basi l'gani but in that ma'amer the Rebbe really opened my eyes to some new
importance to the work that I had been doing in this area. The ma'amer [ ]
about Yisro, the father-in-law of Moshe. The Rebbe earlier in the Ma'amer
discusses the concept Yisro nomi l'choshe--the superiority of light over
darkness. Not just that light is better then darkness but that light by
itself has a certain quality. A certain quality of goodness that when light
is the result of a process of transformation of darkness into light. The
light that comes from darkness is even higher. Its superior. So the Rebbe
mentions that and discusses the concept of Yisro came to Moshe to
acknowledge G- d, Halevaye to be, the Jewish G-d to be greater then all of
the other G-d's, that was the final critical things that had to happen
before G-d could give the Torah to the Jewish people. So the Rebbe wants to
analyze that. Why? A [ ] from King Solomon, Ecclesiastics, I'll read in
Hebrew [hebrew ] I see that there is an advantage of wisdom over
foolishness like the advantage of light over darkness. So we don't need
King Solomon to teach us that wisdom is better then foolishness.[end tape
1]
[tape 2]
Which could mean secular science and or not Torah spirituality. That's my
[premises]. That through the purification and the transformation of wisdom
of the other side to holiness there results in a addition to superiority in
the wisdom of holiness itself. This is why, before the Torah was given,
there had to be the acknowledgement of Yisro. Because Yisro was the
greatest wise man in external wisdoms. The acknowledgement of Yisro that
Havaya, that the Jewish G-d is greater then all the other G-ds this made a
purification in external wisdom so that it became transformed into holy
wisdom, through that there was a superiority and addition in holyism
greater then by itself. So that was on my mind because the ...what I
thought in doing this work was that surely by using mathematical models for
Chassidus would be an elevation for the mathematics itself. That was clear
and obvious. That it makes an elevation in the Torah, that was something
that I could never have conceived of. But that is exactly what the Rebbe
says and he says it over and over again. So that had a tremendous effect on
pushing me harder to work on this stuff and to think of it more and make
time for it and believe also that...and the Rebbe doesn't say this
explicitly in the ma'amer but that Baal Teshuva scientists have a critical
role in bringing Moshiach and the Geulah to take the knowledge that we have
from our scientific work and use it to amplify the Torah in new ways. That
just like Yisro came and he did it and it influenced G-d Al-mighty to give
the Torah and so we have to do it so that G-d Al-mighty will be influences
to bring the Geulah now. Conferences like this, I think are really
important for spreading that. The Rebbe talks in the ma'amer latter after
the quote that I gave that there are two levels in this. The lower level is
where...and he mentions specifically mathematics when mathematics is used
as a model or a mashal for a concept in Torah and adds to our understanding
so then its [hebrew] also it becomes included in Torah. The mathematics
become included in Torah extending the influence of Torah into the world
and bringing the Geula closer. Then the Rebbe says there is even a higher
level where the mathematics actually becomes part of Torah and he gives and
example from the Rambam in Kiddush HaChodesh, in the laws of the
sanctification of the moon, that he uses Greek mathematics in his halachas.
In ruling certain things about the position of the moon and whether certain
testimonies should be accepted by the high court or not. He uses Greek
mathematics, the actual calculations of the Greek mathematicians to decide
halachas in Torah. So that the mathematics is no longer an intermediary
through which are understanding is enhanced but actually becomes a part of
Torah. So [ ] step one here ...but as I already said here, G-d willing Reb
Aryeh Leib will be here and Geulah will be here and I really do think that
when that happens , it should be soon, that we will be able to take step
two and give mathematically precise constructions that will really describe
the seder histalshulos and describe the Creation.
That was
sort of bottom up. That was starting off with the mathematics and then
using it to illuminate a concept in Chassidus. I want to go the other way
now and start of with the Chassidus and then answer an important question
in Chassidus as well as well as a question in mathematics and philosophy.
So the question is, based on the Tzemach Tzedek, and I've been asked this
question by numerous Lubavitchers over the years. One of them was Rabbi
Simon Jacobson and he was the one who helped me work out the answer. He
showed me certain letters from the Rebbe and learned them with me and
explained them to me and this next piece of my talk is basically based on
stuff that Simon learned with me. So the question from the Tzemach Tzedek
its in Derech Mitzvosecha. Its in the ma'amer [052 ] in his discussion of
belief in G-d, so the Tzemach Tzedek makes, he was the third Lubavitcher
Rebbe, he makes a statement that it is impossible that many finite, limited
individuals should join together to form a natural infinity. Its impossible
that many limited individuals should form together to form a natural
infinity. So the question is, how can there be mathematical infinity?
That's precisely what we've done here. We've said that you can take each
one of these finite individuals, the whole numbers and they form together
to form an actual existing infinity. The Tzemach Tzedek says it can't be.
So the main point of the answer is a letter from the Rebbe in response to a
question, actually from Rabbi Altein of Pittsburg. I now have a copy of the
original since I come from Pittsburg and I discussed it with him and he and
I have had many fascinating conversations . Rabbi Altein asked the Rebbe
how do we understand the Tzemach Tzedek saying that it cannot be many
limited individuals should join together to form a natural infinity. This
is a concept of olom [ ] hamispor. Each olom, each world is limited and [ ]
mispor. There are many of them. That's exactly what the Tzemach Tzedek says
can't be. So the Rebbe answers, first of all the fact that there are
infinitely many worlds we have to accept as the truth because the Torah
tells us. That's a given. How do we understand it? There is a basic
principle and one that should be discussed more that G-d creates the world,
if possible, according to our understanding. He creates the world according
to human logic as a general rule which has exceptions, according to the
Rebbe. So the general rule is, the world is created according to our
understanding but He did make some exceptions. Where did he make some
exceptions? In places where the Torah explicitly tells us something that
forces us to say that He made an exception like the fact that He created
infinitely many worlds. So in that case He used koach ain sof that He has.
He took what would normally be impossible and He made it possible. He made
it actual. And He has the ability to do that. So in the case of infinitely
many worlds it is the way it is because the Torah says so and G-d could do
it even though according to our understanding we would say its impossible
and even though normally He doesn't do things in the world that we would
say were impossible. Nonetheless He over ruled the notion of possible in
this case and He made it possible. He made it actual. He created many
worlds. So when the Tzemach Tzedek says that it is impossible that many
limited individuals should join together to form an infinity he is talking
according to the implicit assumption that G- d is going to create the world
according to our understanding. Actually, according to Simon, there is a
very similar concept in Tanya in Sharya Yichud v'Emunah where the Alter
Rebbe says ...where he is proving the concept of hasgacha protise so the
Alter Rebbe says that G-d created the world yesh m'ayin, ex
nihilo--something from nothing like we talked about before and because its
the nature of the physical world not to exist, and the natural thing is for
the world not to exist, so therefore Hashem had to create the world in such
a way that He continually recreates it. So its the same concept. Really
Hashem could have created the world anyway He wanted to. If he wanted to
put the koach into the world that it should continue to sustain itself
after it was created He has the ability to do it. He can do the impossible.
But that would have been more difficult for us to comprehend. The world
doesn't need that koach. It doesn't need that ability because G-d can
recreate the world anyway. Which makes it easier for us to understand it.
So its the same concept. When the Alter Rebbe says its impossible. G-d had
to create the world in such a way that He continues to recreate it, that's
only according to the principle that G-d strives to create the world in
ways that we can understand it according to human logic but that He had the
ability to over ride that also. But we can't say that he did because it
doesn't tell us in the Torah that he did. In this case where the Torah has
told us that He...that there are infinitely many limited worlds, we have to
say that in that case He overruled the notion of impossible and He created
those worlds--infinitely many of them. And that's the answer. I want to
give a little context to what I just said because this question, the
general question of Does there exist and actual infinity? in reality is a
famous question which was discussed by many of the secular geniuses. Many
of them took very firm positions against--Aristotle was one and [Galso] who
was probably the greatest mathematician that ever lived, rejects the
concept of actual infinity. Others say that it could exist or it does
exist. So this is a case where the Torah gives a very definitive answer of
"yes, there does exist an actual infinity." Moreover, the Rebbe,
Shlita, explains the answer in such a way so we can understand did these
geniuses reject an actual infinity. Because they didn't have a notion of
G-d that was great enough to know that He can overcome the impossible. It
really is impossible that there should be an actual infinity. But G-d can
overcome the impossible. But they didn't know that so not only does Torah
give a definitive answer to the question of Does there exist an actual
infinity? The answer is yes. And the Rebbe explains how it could be
and...all in a few words really. So I think this is the best example that I
have of Torah and Moshiach illuminating mathematics and philosophy.
I'm going
to say something a little different to conclude. The spiritual implications
of the mathematical infinity that I have been discussing here are many.
[hebrew] "How great are your creations, Oh G-d" is a fundamental
principle that we have to meditate on. It says in so many places in
Chassidus and the Rambam--contemplating the greatness of G-d leads a person
to proper love of Hashem and so on. So hopefully that already is
enough...people will have a little greater appreciation of that. The other
thing I wanted to mention is the concept of actual infinity existing. The
fact that there is and the fact that for any level of mathematical infinity
there is always a greater one makes more concrete, to me at least, the
concept that however far our own personal service of G-d we have
progresses, we can always go farther. We can always go infinitely farther.
Even in the model here you can see there is little different levels of how
infinitely far you can go. It really is unlimited, how far we can go and
the fact that there are different levels of mathematical infinity makes
more concrete the fact that we can reach a new level which is completely
incomparable to where we were before. So I wrote a paper in a magazine
called Chai Today, in which I ended on that note, a few years ago. And
ended on that note ...unsatisfactory to myself because I felt that I should
give a concrete example and I didn't do it for two reasons: One, I ran out
of pages from the editor and the other reason is I really didn't have such
a good example. But in the last couple of years something happened which
I'd like to share with you which really doesn't have anything directly to
do with mathematical infinity except that it is something that I
experienced from learning a sicha from the Rebbe that really changed my
life in an important way in a critical area.
The sicha
I wanted to discuss if the likud from Parsha Shmos from about two and a
half years ago where the Rebbe talks about the concept of bitochon--of
trust in G-d. I'm not going to pretend to go over the whole sicha.
Basically the concept of trust in G-d is that we should trust in G-d that
if we have a problem, we have a threat, we have a critical situation that
He will help us. He will get us out of that situation in a good way.
Basically the Rebbe describes it as an avodah. An active dynamic service of
Hashem of placing oneself in G-d's hand and truly believing and living with
the positive outcome and the fact that Hashem is doing it and He will help
me. The Rebbe calls it [ an avodah bigo bnafsho] Its work. Its hard labor
of the soul to do this. So it was right around that time that I was endanger
of getting laid off of my job. I was working in a research unit in a
company and we had funding that was disappearing in the next few months.
The chance of new funding was dismal and it was very serious. I'm over
specialized and over qualified in the classic sense. Getting another job is
not that easy especially now with the economy in terrible shape. So this
was pretty serious. In fact people did get laid off and some people were
out of work for a long time before they got another job. This sicha really
absolutely changed my who attitude towards life at that time because
instead of being depressed, which I know would have been the normal
reaction, I was tremendously upbeat. Worked with the sicha and in fact did
get another job from my old boss who I left a few years before and the
detail is I wrote to the Rebbe on my birthday on bais teves which is in the
early winter and I got an answer--a letter signed by the secretary which
came after Pesach. A week later, I spoke to my old boss on the telephone. I
had been talking to him over a period of time. "Yeah, we just go this
major contract. Come right in." So I came right in and by that time we
were just discussing the details of the offer and he says, "Yes, it
was a week ago Friday that we got the official confirmation of this
contract. That was momish the day that the letter arrived over three months
that I wrote to the Rebbe. It wasn't the day before. It wasn't the day
after. So I [ ] the thing really happened and want to conclude with the
notion that with our current situation that bitochon, trust in Hashem is
really a very important concept. And the Rebbe describes in the sicha that
this is the response to make to a crisis which, in some sense we are in,
and that this is something people should be encouraged to work on. To
relearn the sicha. To live with it. And as the Rebbe concludes the sicha,
that in the scus, in the merit, in the bitochon that the geula will happen
it really will happen. And very soon the Rebbe will have a refuah shalama.
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