Translator: Roberto Minelli Reviewer: Michele Gianella
What does ‘unifying’ mean?
How can we build bridges
that allow us to relate different fields and viewpoints
and transfer knowledge from one to another?
In the course of this speech,
I would like to give you my opinion on this subject,
a standpoint I developed over years of research into techniques
to relate and unify distinct mathematical theories.
I have to start by telling you something about my research work,
since that’s the very origin
of this more general insight into the subject of unification.
During my Ph.D. studies, I started to develop a theory
which allows to create bridges between different mathematical theories.
These bridges are based on the fundamental concept of topos,
introduced in the 1960s
by the great mathematician Alexander Grothendieck;
Grothendieck himself had sensed
the unifying potential of this notion,
without, however, using it as a tool
to study the relations among different mathematical theories.
This is exactly what I dedicated myself to,
and over the years I have developed a number of techniques
that enable to efficiently use toposes
as bridges
linking distinct mathematical fields.
You might be wondering, ‘What is a topos?’
It’s difficult not to give a technical answer to this question,
but to a first approximation, you might think of a topos
as an object that embodies
the whole set of standpoints on a given topic.
You may think of it as a place where different perspectives and languages
converge mirroring one another
and, actually, the crucial aspect
of the toposes I use to build these bridges
is the fact that any topos
allows for endless different representations,
which correspond
to the different points of view it embodies.
Thanks to these bridges,
we are actively able to relate –
often in profound and unexpected ways –
notions, properties and results of different mathematical theories
that may well belong to seemingly distant fields
and look disconnected at a first glance.
In other words, they are techniques
that enable us to multiply, in a sense,
standpoints on a given problem
and to discover hidden relations between distinct mathematical contexts.
Now, don’t worry,
I won’t delve into the technical aspects of this research.
What I want to talk about
is instead the general concept of “unifying bridges”,
as inspired by these studies.
By analysing some examples,
graphically illustrated by Silvio Curti, and I thank him for that,
I will try to show you how useful it is
to think in terms of bridges in the most diverse contexts
and also in relation to socially relevant issues
such as integration, enhancement of diversity
and assimilation.
So, let’s start by thinking
about the different approaches we can adopt
to relate different objects or concepts.
When comparing different objects,
we aim to identify invariants.
What is an invariant?
Invariants are the aspects
that objects have in common,
which are not necessarily visible, so to say, to the naked eye,
in a concrete way.
Let’s say that, in some cases,
invariants may be identified through a process of generalisation
we can outline in this way:
in this picture, I have represented, with two dots of different colours,
two concepts we want to compare,
while the black dot
is a more general concept that subsumes the previous two.
Let’s imagine for example
you want to compare human beings to rodents
through a generalisation.
You could point out
that both humans and rodents
are vertebrates or mammals.
This will allow you
to apply to both humans and rodents
all the general informations at your disposal
regarding vertebrates and mammals.
However, you will realise
that this form of unification is not fully satisfying,
because it doesn’t allow for any transfer of information
between rodents and humans.
In fact, you will have noticed that arrows are pointing downwards:
they go from the most general concept to the specific cases,
so they do not enable us to transfer information
between the two objects we want to compare.
We will re-examine this example of humans and rodents
in the light of a more salient form of unification,
which will be provided by a bridge object.
For the time being, what I’d like to point out
is that the reason why a unification based on a generalisation
is static and limited
is that, when generalising, we tend to ignore the specific features
of the objects we want to compare,
to the sole benefit of those that are shared.
In other words,
this kind of unification is boiled down to homogenisation.
Instead, it is legitimate to expect from a profound unification
on the one hand,
to allow for effective transfers of knowledge
between the objects we wish to compare;
and, on the other,
preserve diversity and – possibly – explain its origin.
This kind of unification
implies the existence of a bridge object.
What is a bridge object?
The aim is to relate
our two objects to each other
thanks to a third entity – a third object
that will, precisely, be our bridge object,
which can be linked to each of the two entities separately.
Now, this bridge object may have a nature
that is completely different from that of the two entities;
there are no restrictions about it.
The only thing that matters is that one should be able to build,
though not necessarily in a concrete fashion,
the same object
from the two entities taken individually.
This object will act as a bridge
to transfer knowledge from one object to the other.
How?
How do these transfer of knowledge work?
By analysing the way properties of the bridge object
are expressed in each of the two distinct entities
upon which the bridge object is built.
Each property of the bridge object will originate an invariant,
whose expression is unique with respect to the bridge object,
but whose concrete manifestations in the two entities
can be completely different.
Just think, for instance, of a feeling: it can be expressed in writing,
through a gesture,
or even through music or words.
As you can see, an invariant may be expressed
in many different ways.
You can see that this kind of bridge-based unification
allows to transfer knowledge
from one object to the other as indicated in the picture,
as opposed to what happened
when the unification was based on a generalisation.
In fact, reverting to the example of rodents and humans,
we can see that by properly identifying a bridge object,
we get to achieve a much more substantial, far more useful unification
between rodents and humans
than those we might obtain
through simple generalisations.
A bridge object that naturally comes to mind to consider,
if we aim to study the relations between human beings and rodents,
is the part of DNA they share.
This indeed can play the crucial role of bridge object.
In fact, in spite of being an invariant, this shared DNA – an invariant –
expresses itself in humans and rodents
in completely different ways.
As a matter of fact, the depth of this form of unification,
in contrast with those given by generalizations,
is proven by the great success of the scientific studies on rodents
that were carried out in the last decades
to gain a better understanding of a big deal of diseases.
Please notice that,
while a simple generalization
didn’t take you that far as it was impossible, for example,
to treat a disease by studying rodents’ physiology,
in this case we can transfer information
from rodents to humans
and this, obviously, has significant consequences.
Therefore, you can easily notice that while generalization
led to a unification based on homogenisation,
bridges allow to create a unification
where diversity isn’t diluted into abstraction,
but rather enhances it,
because the diversity of the single objects
determines the different forms
in which invariants, defined at the bridge object’s level,
express themselves.
So, in other words, we have a sort of morphogenesis
which explains the origin of the diversity
of different expressions of the same invariant.
Another natural example of bridge, drawn from astronomy,
is given by planets revolving around the same star,
such as the planets of the solar system.
If you think about it, in this instance the Sun is the bridge object
that causes
all planets revolving around it
to have elliptic orbits
that, in spite of their diversities,
share a number of features.
As a matter of fact, Kepler’s laws show us that the invariants,
which are the aspects shared among the different orbits
of the different planets,
live abstractly at the level of the Sun,
but manifest themselves in different ways in the context of different planets.
Historically speaking, ignoring these invariants
in favour of tangible viewpoints, like those of the planets,
instead of adopting the viewpoint of the bridge object, the Sun,
as for example in the Ptolemaic system,
caused significant problems when describing their motions.
The Copernican revolution suggested a change in perspective
by shifting the focus from Earth to the Sun.
This led to a number of important developments;
in other words,
the identification of the bridge object
resulted in significant progress in terms of knowledge.
Let’s now switch to another context, a linguistic one,
where bridges are easily found:
the translation of texts from one language to another.
If you try to produce a literal translation
strictly based on the use of a dictionary,
you’ll find yourself in a perspective of homogenisation,
which you will soon realise is an approach
that – if at all possible,
because should the languages be too different from one another,
such an approach clearly cannot work
since a given meaning will be expressed
in too syntactically different a way, in two languages –
for a literal translation to take place –
but even if it were possible,
it would hardly be satisfying.
So, how can we produce a good translation?
First of all, we should identify the invariants,
which is what we want to remain unchanged – preserved –
in the transition from one language to the other.
Of course, among these invariants,
the most important one is obviously meaning,
but there are others too.
For instance, when you translate a poem, you might want to preserve other aspects,
such as a particular type of metre or musicality.
Anyway, whatever the invariants,
what matters is to focus on them while doing the translation,
to translate in the light of these invariants,
which will be our bridge objects.
In other words, we should consider
how these invariants express themselves in the two languages.
If you think about it, this is exactly
what we do, as a rule unconsciously,
when we work hard to translate intelligently.
This is what I do in mathematics too,
when I transfer notions and results from one theory to another:
in my case the objects to be related
are mathematical theories
and the bridge objects I use are toposes associated to them,
on which an infinite number of invariants are defined.
If you still have in mind the image of planets and the Sun,
you could represent the toposes
as suns that enlighten mathematical reality,
as universal standpoints on theories
which naturally lets you identify their symmetries.
To sum up, the very concept of bridge shows that, in general, two objects
share many more features
than we might be led to expect.
In order to discover and understand these relations
it is often necessary to take a perspective
that’s different from the one given by the objects themselves:
that of a bridge object that can connect them.
That’s because the invariants
that we can see, so to speak, with the naked eye,
or without changing point of view,
are relatively few
and, above all, aren’t necessarily the most relevant.
It is therefore important to bear in mind
that a bridge object can have a completely different nature
from that of the entities we want to compare.
More specifically, it could be decidedly abstract
like a topos associated with a theory
or ineffable, like the meaning of a sentence,
or very far, like a star a planet revolves around.
It is thus essential to consider two levels:
the level of the entities we want to compare
and the one of the bridge objects that can connect them,
thus clearly distinguishing the invariants’ level,
defined at the level of the bridge object,
and the one of their actual manifestation
in the context of the different entities.
To see the invariants we have to think in abstract terms:
there’s no other way.
If we want to build bridges,
we must be inspired by concepts
that can’t be reduced to a particular manifestation.
Think about ideals like beauty, rationality, justice and love:
yes, they are all partially unclear concepts,
but their strength lies in their very abstraction,
because they are bridge concepts.
With their huge amount of different manifestations,
both actual and potential,
these concepts inspire us
and teach us we should never box ourselves
into the partiality of a single standpoint, or experience,
but should always reach beyond seeking new points of view
that allow us to discover hidden relations
with things and people around us.
Thank you.
(Applause)
What does ‘unifying’ mean?
How can we build bridges
that allow us to relate different fields and viewpoints
and transfer knowledge from one to another?
In the course of this speech,
I would like to give you my opinion on this subject,
a standpoint I developed over years of research into techniques
to relate and unify distinct mathematical theories.
I have to start by telling you something about my research work,
since that’s the very origin
of this more general insight into the subject of unification.
During my Ph.D. studies, I started to develop a theory
which allows to create bridges between different mathematical theories.
These bridges are based on the fundamental concept of topos,
introduced in the 1960s
by the great mathematician Alexander Grothendieck;
Grothendieck himself had sensed
the unifying potential of this notion,
without, however, using it as a tool
to study the relations among different mathematical theories.
This is exactly what I dedicated myself to,
and over the years I have developed a number of techniques
that enable to efficiently use toposes
as bridges
linking distinct mathematical fields.
You might be wondering, ‘What is a topos?’
It’s difficult not to give a technical answer to this question,
but to a first approximation, you might think of a topos
as an object that embodies
the whole set of standpoints on a given topic.
You may think of it as a place where different perspectives and languages
converge mirroring one another
and, actually, the crucial aspect
of the toposes I use to build these bridges
is the fact that any topos
allows for endless different representations,
which correspond
to the different points of view it embodies.
Thanks to these bridges,
we are actively able to relate –
often in profound and unexpected ways –
notions, properties and results of different mathematical theories
that may well belong to seemingly distant fields
and look disconnected at a first glance.
In other words, they are techniques
that enable us to multiply, in a sense,
standpoints on a given problem
and to discover hidden relations between distinct mathematical contexts.
Now, don’t worry,
I won’t delve into the technical aspects of this research.
What I want to talk about
is instead the general concept of “unifying bridges”,
as inspired by these studies.
By analysing some examples,
graphically illustrated by Silvio Curti, and I thank him for that,
I will try to show you how useful it is
to think in terms of bridges in the most diverse contexts
and also in relation to socially relevant issues
such as integration, enhancement of diversity
and assimilation.
So, let’s start by thinking
about the different approaches we can adopt
to relate different objects or concepts.
When comparing different objects,
we aim to identify invariants.
What is an invariant?
Invariants are the aspects
that objects have in common,
which are not necessarily visible, so to say, to the naked eye,
in a concrete way.
Let’s say that, in some cases,
invariants may be identified through a process of generalisation
we can outline in this way:
in this picture, I have represented, with two dots of different colours,
two concepts we want to compare,
while the black dot
is a more general concept that subsumes the previous two.
Let’s imagine for example
you want to compare human beings to rodents
through a generalisation.
You could point out
that both humans and rodents
are vertebrates or mammals.
This will allow you
to apply to both humans and rodents
all the general informations at your disposal
regarding vertebrates and mammals.
However, you will realise
that this form of unification is not fully satisfying,
because it doesn’t allow for any transfer of information
between rodents and humans.
In fact, you will have noticed that arrows are pointing downwards:
they go from the most general concept to the specific cases,
so they do not enable us to transfer information
between the two objects we want to compare.
We will re-examine this example of humans and rodents
in the light of a more salient form of unification,
which will be provided by a bridge object.
For the time being, what I’d like to point out
is that the reason why a unification based on a generalisation
is static and limited
is that, when generalising, we tend to ignore the specific features
of the objects we want to compare,
to the sole benefit of those that are shared.
In other words,
this kind of unification is boiled down to homogenisation.
Instead, it is legitimate to expect from a profound unification
on the one hand,
to allow for effective transfers of knowledge
between the objects we wish to compare;
and, on the other,
preserve diversity and – possibly – explain its origin.
This kind of unification
implies the existence of a bridge object.
What is a bridge object?
The aim is to relate
our two objects to each other
thanks to a third entity – a third object
that will, precisely, be our bridge object,
which can be linked to each of the two entities separately.
Now, this bridge object may have a nature
that is completely different from that of the two entities;
there are no restrictions about it.
The only thing that matters is that one should be able to build,
though not necessarily in a concrete fashion,
the same object
from the two entities taken individually.
This object will act as a bridge
to transfer knowledge from one object to the other.
How?
How do these transfer of knowledge work?
By analysing the way properties of the bridge object
are expressed in each of the two distinct entities
upon which the bridge object is built.
Each property of the bridge object will originate an invariant,
whose expression is unique with respect to the bridge object,
but whose concrete manifestations in the two entities
can be completely different.
Just think, for instance, of a feeling: it can be expressed in writing,
through a gesture,
or even through music or words.
As you can see, an invariant may be expressed
in many different ways.
You can see that this kind of bridge-based unification
allows to transfer knowledge
from one object to the other as indicated in the picture,
as opposed to what happened
when the unification was based on a generalisation.
In fact, reverting to the example of rodents and humans,
we can see that by properly identifying a bridge object,
we get to achieve a much more substantial, far more useful unification
between rodents and humans
than those we might obtain
through simple generalisations.
A bridge object that naturally comes to mind to consider,
if we aim to study the relations between human beings and rodents,
is the part of DNA they share.
This indeed can play the crucial role of bridge object.
In fact, in spite of being an invariant, this shared DNA – an invariant –
expresses itself in humans and rodents
in completely different ways.
As a matter of fact, the depth of this form of unification,
in contrast with those given by generalizations,
is proven by the great success of the scientific studies on rodents
that were carried out in the last decades
to gain a better understanding of a big deal of diseases.
Please notice that,
while a simple generalization
didn’t take you that far as it was impossible, for example,
to treat a disease by studying rodents’ physiology,
in this case we can transfer information
from rodents to humans
and this, obviously, has significant consequences.
Therefore, you can easily notice that while generalization
led to a unification based on homogenisation,
bridges allow to create a unification
where diversity isn’t diluted into abstraction,
but rather enhances it,
because the diversity of the single objects
determines the different forms
in which invariants, defined at the bridge object’s level,
express themselves.
So, in other words, we have a sort of morphogenesis
which explains the origin of the diversity
of different expressions of the same invariant.
Another natural example of bridge, drawn from astronomy,
is given by planets revolving around the same star,
such as the planets of the solar system.
If you think about it, in this instance the Sun is the bridge object
that causes
all planets revolving around it
to have elliptic orbits
that, in spite of their diversities,
share a number of features.
As a matter of fact, Kepler’s laws show us that the invariants,
which are the aspects shared among the different orbits
of the different planets,
live abstractly at the level of the Sun,
but manifest themselves in different ways in the context of different planets.
Historically speaking, ignoring these invariants
in favour of tangible viewpoints, like those of the planets,
instead of adopting the viewpoint of the bridge object, the Sun,
as for example in the Ptolemaic system,
caused significant problems when describing their motions.
The Copernican revolution suggested a change in perspective
by shifting the focus from Earth to the Sun.
This led to a number of important developments;
in other words,
the identification of the bridge object
resulted in significant progress in terms of knowledge.
Let’s now switch to another context, a linguistic one,
where bridges are easily found:
the translation of texts from one language to another.
If you try to produce a literal translation
strictly based on the use of a dictionary,
you’ll find yourself in a perspective of homogenisation,
which you will soon realise is an approach
that – if at all possible,
because should the languages be too different from one another,
such an approach clearly cannot work
since a given meaning will be expressed
in too syntactically different a way, in two languages –
for a literal translation to take place –
but even if it were possible,
it would hardly be satisfying.
So, how can we produce a good translation?
First of all, we should identify the invariants,
which is what we want to remain unchanged – preserved –
in the transition from one language to the other.
Of course, among these invariants,
the most important one is obviously meaning,
but there are others too.
For instance, when you translate a poem, you might want to preserve other aspects,
such as a particular type of metre or musicality.
Anyway, whatever the invariants,
what matters is to focus on them while doing the translation,
to translate in the light of these invariants,
which will be our bridge objects.
In other words, we should consider
how these invariants express themselves in the two languages.
If you think about it, this is exactly
what we do, as a rule unconsciously,
when we work hard to translate intelligently.
This is what I do in mathematics too,
when I transfer notions and results from one theory to another:
in my case the objects to be related
are mathematical theories
and the bridge objects I use are toposes associated to them,
on which an infinite number of invariants are defined.
If you still have in mind the image of planets and the Sun,
you could represent the toposes
as suns that enlighten mathematical reality,
as universal standpoints on theories
which naturally lets you identify their symmetries.
To sum up, the very concept of bridge shows that, in general, two objects
share many more features
than we might be led to expect.
In order to discover and understand these relations
it is often necessary to take a perspective
that’s different from the one given by the objects themselves:
that of a bridge object that can connect them.
That’s because the invariants
that we can see, so to speak, with the naked eye,
or without changing point of view,
are relatively few
and, above all, aren’t necessarily the most relevant.
It is therefore important to bear in mind
that a bridge object can have a completely different nature
from that of the entities we want to compare.
More specifically, it could be decidedly abstract
like a topos associated with a theory
or ineffable, like the meaning of a sentence,
or very far, like a star a planet revolves around.
It is thus essential to consider two levels:
the level of the entities we want to compare
and the one of the bridge objects that can connect them,
thus clearly distinguishing the invariants’ level,
defined at the level of the bridge object,
and the one of their actual manifestation
in the context of the different entities.
To see the invariants we have to think in abstract terms:
there’s no other way.
If we want to build bridges,
we must be inspired by concepts
that can’t be reduced to a particular manifestation.
Think about ideals like beauty, rationality, justice and love:
yes, they are all partially unclear concepts,
but their strength lies in their very abstraction,
because they are bridge concepts.
With their huge amount of different manifestations,
both actual and potential,
these concepts inspire us
and teach us we should never box ourselves
into the partiality of a single standpoint, or experience,
but should always reach beyond seeking new points of view
that allow us to discover hidden relations
with things and people around us.
Thank you.
(Applause)
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