# Objectivity and Subjectivity in Science and Religion: Towards Basing Inter-Religious Dialogue on Rational Grounds

Is objectivity a characteristic of scientific knowledge?

What is the root of the objective and public nature of science?

Mathematics as an objective nucleus of scientific knowledge.

What characterises mathematics as an objective nucleus of science?

Is religion something subjective and based on subjective and private experiences in contrast with the objective and public nature of science?

Can we formalise religious language in a way similar to what we have done with the language of science?

What similarities and differences are there between the objective nucleus of science and the objective nucleus of religions?

**A. Science and objective knowledge**

1. Is objectivity a characteristic of scientific knowledge?

Although scientific knowledge is never completely objective and independent of the subject, the claim to be objective knowledge characterises science.

Objectivity & communication: The objectivity of scientific knowledge enables two distinct, unrelated observers to have the capacity to obtain scientific data which falsify the same scientific proposition.

Objectivity & Universality: The objectivity of scientific knowledge means that scientific knowledge is somehow independent of the cultural context in which it develops. Science is the same in all cultures and is a unifying component of the diversity of cultures in which scientific activity takes place. The objective exists independently of our consciousness. Knowledge is objective when it is transmittable from a knower subject to another without being modified by the fact of the transmission.

Objectivity & Technology: The objectivity of scientific knowledge makes it possible to implement scientific knowledge in technological instruments which can be handled by persons regardless of their personal and cultural characteristics.

2. What is the root of the objective and public nature of science?

There is a direct relation between the mathematical formulation of scientific knowledge and its objective value.

Mathematics vs. Scientific Paradigmes.

Different sciences use different types of models to represent reality (physical, biological, neuro-scientific etc.). The objectivity of these models is restricted by the paradigm which is proper to the different sciences. These are models which must be understood within the context of a science.

Contrary to this, the formal semantic models of mathematics claim to be valid in the context of all the sciences. These formal semantic models are characterised by their objectivity and by the formal clarity with which they are defined. Due to their objective and public nature, these models enable men and women to have a common vision of scientific reality regardless of their religions, races or cultures. The mathematical models are used to obtain an objective and public understanding of the models of the different sciences.

Mathematical knowledge vs Scientifical knowledge.

I distinguish between two types of scientific knowledge: empirical and mathematical. There are profound interrelations and substantial differences between empirical knowledge and mathematical knowledge. I wish to point out two differences and one profound interrelation. The first difference is that empirical knowledge is hypothetical, that is to say, it is based on hypotheses which can be shown to be wrong by observations, while mathematical knowledge cannot be shown to be wrong by empirical experience. The second difference will be shown by distinguishing between the formal models which satisfy the mathematical theories and the representative models which explain the empirical theories. Despite these two differences, there is a relationship between the two types of knowledge as we only totally understand and control that part of empirical knowledge which we have been able to express as mathematical knowledge.

First difference between empirical and mathematical knowledge: The falsifiable nature of hypothetical-empirical propositions and the non-falsifiable nature of axiomatic-mathematical propositions

The first difference between empirical and mathematical knowledge lies in the same nature of the basic propositions of both types of knowledge. The basic propositions of empirical sciences are hypotheses built around certain particular observations and, thus, can be shown to be false through further experiments and observations while the basic propositions of mathematics are axioms based on mathematical intuitions applicable to any observation and, therefore, they are not shown to be false by new experiments and observations.

Examples of mathematical axioms:

Some axioms of the theory of sets of Zermelo Fraenkel:

“There is a set with no elements”

“There is a set with two elements”

“There is a set made up of the union of all the sets of a given set”.

Some axioms of Euclid’s geometry:

- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefintely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
- All right angles are congruent.
- If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must inevitably intersect each other on that side if extended far enough. A briefer and equivalent way to state this proposition is given by Proclus (411-485): “Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line”.

Examples of empirical hypotheses:

A hypothesis of quantum mechanics: “Energy is exchanged in quanta discretely”

A hypothesis of classical mechanics: “Energy is exchanged continuously”

An indeterminist quantum hypothesis: “It is impossible to fix the position and the time of a particle at the same time”

A classical determinist hypothesis: “It is possible to fix the position and the time of a particle”

The Ptolemaic hypothesis: “The planets gravitate round the earth”

The Copernican hypothesis: “The planets gravitate round the sun”

Historical revolutions of empirical science and the historical evolution of mathematics.

The nature of scientific hypotheses has meant that science has undergone historical revolutions due to the fact that the hypotheses on which science is based were shown to be false. However, mathematics has grown, has changed and has evolved throughout history in a different way. When empirical hypotheses are shown to be false, this leads to scientific revolutions, while the evolution of mathematics can do without previous axioms, but it does not necessary affirm them to be false. For example, Newtonian physical laws which describe the trajectories, the position, and speed of a particle were shown to be false by new relativistic observations and the Ptolemaic hypothesis that all the planets gravitate around the earth was shown to be false by experimental observations which contradict this hypothesis. However, the arithmetical and geometrical theorems regarding the point, the straight line, the zero, the number two cannot be shown to be false by other observations. I cannot make an observation which shows that 2 + 2 = 4 is false. I can accept the mathematical axioms or not accept them, but I cannot empirically demonstrate that they are false. I can accept the fifth axiom of Euclid or I can reject it and develop a non-Euclidian geometry, but I cannot demonstrate by empirical observation that the fifth axiom of Euclid is false.

Second difference between empirical and mathematical knowledge: The representative models and semantic-formal models.

The experimental scientist constructs representative models which metaphorically reproduce the conduct of reality under certain hypotheses. The propositions on these representative models can be shown to be false through new observations. The mathematician, on the other hand, uses a different kind of models. She uses formal semantic models which interpret the meaning of formal propositions through a clear univocal correspondence. Given a formal proposition and a semantic model the mathematician will try to proof the truth or the falsehood of the proposition, if the mathematician is not able to proof the truth or falsehood, he will look for new formal proofs but will not have recourse to an empirical experiment in order to show that the proposition is false.

Examples of representative models:

Two examples of representative models are the Bohr’s atom and the Rutherford’s atom models. Bohr’s atom model exemplifies the hypotheses of quantum leaps, which are proper to quantum mechanics, while Rutherford’s atom exemplifies the hypotheses of classical mechanics. Quantum leaps are represented in the model of Bohr’s atom through electron leaps between the orbits, while the movements of classical mechanics are represented in Rutherford’s atom by classical electron orbits. The leaps between orbits of electrons in Bohr’s atom metaphorically represent the observations of quantum leaps. The representative models of Rutherford and of Bohr metaphorically represent the reality in agreement with different scientific visions in each case.

Example of a formal-semantic model:

One example of a formal model is the structure <N, sN> where N={0,1,2,3… } is the set of natural numbers and sN is the function defined on the set N as a correspondence between each natural number x to its successor sN(x): sN(0)=1, sN(1)=2, sN(2)=3…

Given the following two mathematical propositions on natural numbers in which the successor function appears:

For every natural number x there is a natural number y such that y is a successor of x

There is a natural number x which is successor of itself,

We can formally write these propositions as formulas of first order logic:

j1 º "x $ y (y = s(x))

j2 º $ x (x = s(x))

Once they are formally written, we can check that the formula j1 is true in the structure <N, sN> and that the formula j2 is false in <N, sN>. In fact, j1 is true because given any number 0,1,2… of the set N, we can correspond it to its successor through the function sN, however, j2 is false in <N, sN> because there is no number of N which is a successor of itself.

Proof of the truth or falseness of a proposition in a formal model.

We state that j1 is true in <N, sN> and that j2 is false in <N, sN> not as a consequence of an empirical observation but as a consequence of a formal proof. The first step of this formal proof consisted of defining a correspondence between the sign s, which appears in the two formal propositions j1, j2, and the successor function sN of the formal semantic model <N, sN>. The second step of the proof consisted in showing that, in accordance with this correspondence, <N, sN> satisfies j1 and it does not satisfies j2, that is to say, j1 is true in <N, sN> and j2 is not true in <N, sN>.

The metaphorical representative models are linked to a community and to a paradigm.

We have seen that the models in the empirical sciences are frequently metaphorical. The metaphorical scientific descriptions are understood in the context of a certain scientific paradigm. The metaphorical language is proper to a certain scientific community which has accepted a certain scientific paradigm and its representative models are understood and accepted only within this scientific community. When a scientific community creates a new paradigm, it uses metaphors to explain the new paradigm. For example, Bohr’s atom explains quantum leaps using the metaphor of the orbits of the planets.

The formal language of semantic models is trans-paradigmatic and trans-cultural.

On the contrary, the formal descriptions of semantic models are totally trans-cultural and trans-paradigmatic. The formal entities have the same meaning for all communities. The formal entities are totally public; they can be translated to any cultural context. They are not expressed by means of metaphorical images.

Interrelation between empirical and mathematical knowledge.

We have seen that the empirical sciences and mathematics are differentiated by their different relationship to the empirical perceptions of the real world. However, despite the fact that the empirical sciences and mathematics deal with two types of different knowledge, we cannot separate them. We discover mathematical structures in the perceptions of the real world and we scientifically describe empirical perceptions with the aid of mathematical structures. For example, in order to describe the position and the trajectory of an elementary particle of matter, we use the mathematical intuitions of the point, the straight line, the circle, etc. and in order to quantify the speed and the time of the particle, we use the mathematical concepts of the zero, the natural numbers, the addition, the product of two numbers, etc.

Mathematics is the formal nucleus which provides structure to empirical theories.

Mathematics is like the skeleton of the body. There are parts of scientific theory which are described in formal mathematical language and which are supported in basic mathematical concepts and parts which are described through a non-formal metaphorical language. The non-formal part of the scientific theories is the part which gives a hypothetical character to knowledge. For example, in Heisenberg’s quantum principle of uncertainty, the mathematical formula Dx · Dp ≥ h/4π describes the relationship between the deviation of the position of a particle Dx, the deviation of the time Dp and the constant of Planck h. The values of x, and p represent measures of observations and the formula can be shown to be false by further observations. However, Dx · Dp ≥ h/4π is a mathematical formula which establishes a relationship between the increase of two variables Dx, Dp and certain constant values h, 4 and π in a numerical model. The numerical relationship Dx · Dp ≥ h/4π is independent of the observations and is a formal structure of the principle of indeterminacy.

The objectiveness of scientific knowledge is based on the existence of a formal mathematical nucleus.

The mathematical formulation of scientific knowledge makes it possible for two unrelated, totally independent observers to accept or reject the same scientific hypothesis. The mathematical formulation of the scientific hypotheses makes these hypothesis independent from of their interpretation within a particular paradigm.

3. Mathematics as the objective nucleus of scientific knowledge

Throughout history, different scientific paradigms have appeared. Different paradigms frequently correspond to different scientific and cultural times. Different paradigms admit often contradictory hypothesis as valid. For example, the astronomic model of the Ptolemeic system admits as valid that the earth is the centre of the planetary system while the Copernican model does not admit this, and, towards the end of the XIX century, scientists thought that there was a continuous medium called the ether which filled all space while this is not admitted at the present time.

However, despite the diversity of scientific interpretations of the same facts, there is an objective nucleus in science. Scientific knowledge is public and the propositions of Ptolemeic astronomy can be studied at the present time, studied and partially accepted or rejected. They may be consi ered to be false but they continue to be objective.

The mathematical formulation represents the objective nucleus of the scientific results. The mathematical formalisms reach the maximum degree of objectivity of scientific knowledge. Mathematical certainty is objective certainty because it is a certainty which is transmissible through the objectivity of mathematical language. Once a scientific proposition is formally formulated, it can be falsified.

Mathematics is a formal instrument common to all the sciences, which makes sciences objective because the propositions formally formulated can be falsified.4. What characterises mathematics as an objective nucleus of science?

Mathematics has evolved throughout history. Along the centuries, math has laboriously obtained a corpus of clearly valid statements by inferring their objective certainty from simple basic intuitions. Among these basic intuitions the ones of Arithmetic and Geometry are of special importance. Mathematics represents arithmetical and geometrical objects by means of elementary signs associated to numbers and figures and then enunciates valid sentences about these objects.

The mathematics of the XIX century is based on the analysis of infinite sets of numbers. The discovery of paradoxes in the language of mathematics led to a crisis of certainty and the objectivity of mathematical knowledge and, consequently, of scientific knowledge.

The formalism of Hilbert was the most serious attempt to establish the objectivity of mathematical knowledge. For formalism, as understood by Hilbert, there is a privileged part of mathematics which is based on the pure intuition of specific and discrete signs. Hilbert intended to demonstrate the objective and certain nature of classical mathematics by finite, sure means, from certain specific formal signs which we have direct intuition of.

While the objectivisation of the different empirical models of the different sciences does not manage to become total and certain, we achieve a greater degree of objetivisation when we can reduce the scientific models to simple discrete components such as numbers, sets, and geometrical figures. These basic discrete components are objects we have an immediate intuitive experience of. The objectivity of these basic components means that we can precisely delimit whether they are equal to or different from each other, what basic components constitute them, or whether they are not constituted by a more basic component. The meaning of these basic objects is perceived immediately and is the same for all. Everyone can univocally understand what the number one is and what the successor function involving the passage of one number to its successor.

5. Scientific objective knowledge enables technological action

Technological action is the handling and use of nature in accordance with objective mathematical knowledge. There are two fundamental dimensions of technology which correspond to the two types of knowledge on which they are based.

There is a dimension of technology which is based on knowledge of the laws formulated mathematically and is totally objective and neutral as regards philosophical and cultural viewpoints. I will call this part formal-mechanical technology. There is another dimension of technology based on scientific hypotheses. I will call this part empirical-scientific technology.

Formal-mechanical technology.

The mathematical formulation of scientific theories makes it possible for us to implement these technologically. Machines only understand mechanical instructions. For example, when I go down gear in a car, I am transmitting mechanical information to the car so that it will function with higher revolutions. The mechanical instructions can be formalised by translating them to formal algorithms and the formal instructions can be implemented mechanically through a succession of mechanical actions. The most significant example of the equivalence between mechanical instructions and formal algorithms are the Turing machines. The Turing machines are formal mechanisms which can represent the action of any computer which can be built. They were first described by Alan Turing in 1936. Formal-mechanical technology has had an exponential explosion with the development of computing systems.

Empirical-scientific technology.

If a computer is to function, it requires software, formal programs and computing systems, as well as micro-processor hardware and transistors. The software is a product of formal-mechanical technology, based on formal-mechanical rationality. The hardware is a product of empirical-scientific, based on hypothetical-empirical knowledge. Formal mathematics makes empirical scientific knowledge objective. The objectivity of mathematical formulations enables scientific theories to be implemented by technological instruments. The technological instruments are based on formal theories and, as functioning, are independent of any paradigm or cultural context. Despite the diversity of the scientific interpretation of the same facts through different paradigms, the technological instruments, based on mathematical formulations continue to function in the same way. The same images of the same telescope can be interpreted through Newtonian mechanics or through relativist mechanics.

We only technologically control that part of empirical knowledge which we have been able to express as mathematical knowledge.

We have seen that we can only express part of empirical knowledge as mathematical knowledge. We have also seen that machine act under blind formal orders. The consequence is that we can only transmit to machines the part of scientific knowledge which is formulated mathematically.

6. Technological action is essentially open to risk and innovation.

Openness to risk and innovation has three levels. 1) At the most basic level, the formal-mechanical technology is open to risk and innovation due to the very nature of mathematics, as the mathematical theorems of incompleteness and undecidability show that we cannot decide on the result of mathematical calculations in all the cases. We cannot execute any algorithm without risking in some cases the result. 2) Risk acquires a second, deeper dimension in empirical-scientific technology due to the hypothetical nature of empirical knowledge. 3) However, the more substantial openness to risk and innovation for technological action comes from the fact that technology is applied science. Technology is action and as human action it includes human options.

As technological action is open to risk and innovation, it has the free human subject who chooses as protagonist.

Technology is action on the world essentially open to risk and innovation. Science studies the world as it is. Technological action attempts to construct the world as the agent wishes it to be. Technological action is carried out by the individual as a member of a particular scientific community and of a society. Technology was created by social interaction, it transforms society and creates new meanings for human interaction.

7. Technological action needs to be based on options with meaning.

Technological action transforms the human being and society, it creates new relationships of the human being with nature and with society. Technological action needs to have a rational meaning. Technological action has a rational meaning when it is based on an option which is suited to an image of nature, of the human being and of society. That is to say, when it corresponds to a cosmic vision, a philosophy of life, a culture or a religion.

It is natural for a human being to base the meaning of life in a cosmic-vision.

Technological action does not have a criteria of meaning in itself. By itself, and with no external meaning, technological action tends to grow like a cancer without responding to the global needs of nature, the human being and society. It creates necessities which lack meaning. Throughout history meaning has been differently understood in religions, ideologies, philosophies, cultures…

B. Religion and objective knowledge

- Is the meaning, as it is understood in religions, cultures and philosophies, something subjective based on subjective and private experiences in contrast with the objective and public nature of science?
- Are in particular religious experiences subjective and particular or are they public and communicable to any person, regardless of his culture, race or gender?

Religious communication supposes religious universality and religious universality supposes that the content of faith is public and that it is possible for believers from different religions and non-believers to speak about this. In many religions, and in Christianity in particular, there is the claim that religious faith is not something merely subjective. The inter-religious dialogue is based on the objective value which religious propositions might have, regardless of the individual sentiments and perceptions of the subject speaking.

Similarly as in the case of empirical sciences, knowledge of meaning and religious faith both are also based on a specific kind of experience that we will call religious faith experience. Religious faith experiences are also formalized in the form of religious writings, Scriptures, traditional texts, dogmas, etc. There are clear interrelations and substantial differences between empirical scientific experience and religious experiences. Similarly as in the case of science faith experiences need to be confronted with communitarian religious formulations and confirmed by other experiences. The prototype example of religious experience among Christians is the experience of the Resurrection of the Lord. When the disciples had the experience of the apparition of the Lord they where confirmed in the faith and they had the need to transmit their experience to the community of believers.

10. Can we formalise religious language in the same way as we have formalised the language of science?

We have seen that in the case of scientific knowledge the characteristic of being objective and public is based on the capacity to formally express the content of science. Scientific formalism is based on the direct intuition of certain basic signs which can be perceived by all. These basic signs are equal for all and everyone perceives them in the same way. From basic propositions concerning these signs, we can deduce a certain, sure and objective part of mathematics. For this formalisation to have an objectifying character, as in the case of science, it would have to refer to basic concepts which might be indistinctly perceived by everyone.

What is the objective nucleus of religious knowledge and in what sense can we say that this nucleus is objective?

The spiritual religious experience is objective in the sense that different people from different cultural and spiritual traditions can communicate about their spiritual experience as it is formalized in the form of texts and writings. In this sense the religious experience is not totally subjective and conditioned by the subject who experiments it. Religious experience, as it is formalized in a religious tradition, constitutes the objective nucleus of religious knowledge.

What similarities and differences are there between the objective nucleus of science and the objective nucleus of religions?

Formal finitist mathematics forms a nucleus common to the multiplicity of manifestations of science. The formal finitist intuitions are present in all the scientific models. The scientific models can be contradictory with each other, but they all coincide in the common nucleus of formal finitist mathematics.

Formal finitist mathematics is based on the intuition of the significance of certain basic formal signs which are the subjects of finitary reasoning. From this intuition, a set of finitary propositions is formulated. Then methods of finitary reasoning are used in order to obtain new propositions.

Religious spiritual experience is the nucleus common to the multiplicity of manifestations of religions. Religious experiences are formulated differently in the different the religions. The different religions may be contradictory with each other in their formulations, but they all have something experiential in common.

The basic differences between the objective nucleus of science and the objective nucleus of religions lies in the fact that the difference between the cognitive experience of the basic signs of mathematics and the cognitive experience of the basic religious symbols. Mathematical signs are essentially univocal and refer exclusively to the formal object perceived. The cognitive spiritual experience is also simple and unique, but it is also global and simultaneously refers to all the objects of experience.

Towards a grounding of the inter-religious communication on a rational basis

The study of the religious experience provides support for grounding the inter-religious communication on a rational basis because it presents a common platform for communication which can be accepted by all.

An inter-religious dialogue based on the religious experience would start from a rational analysis of the religious experience, by formulating a set of propositions on the religious experience which might be admitted by all people involved in the dialogue, and would take these propositions as the grounds for their dialogue.

Paradoxical relation between formal and religious reasoning

In some aspects, formal and religious reasoning are opposed:

- Formal signs are opposed to religious symbols

Signs: Mathematical propositions refer to signs with concrete defined meanings.

Symbols: Religious and theological propositions refer to symbols whose meanings refer to the whole of reality and the transcendent God.

- Ineffability is opposed to Communicability.

Ineffability: Many religious people feel they have a linguistic incapacity to communicate the positive experience of the ultimate meaning of the world and God.

Communicability: Formal mathematical propositions are communicable to any person. - Passivity is opposed to Control

Passivity: Religious visions of the world do not permit a positive control of reality.

Control: Technological control of reality is a consequence of the formal knowledge of the laws of nature. - Holism is opposed to Definition

Holism: Religious propositions refer to the totality of reality.

Definition: Mathematical formulas always refer to a particular domain of discourse. - Simplicity is opposed to Complexity

Simplicity: Religious knowledge of what is global is synthetic and simple.

Complexity: formal knowledge is analytical and complex.

Paradoxically, formal reasoning is also:

- Simple, because it does not change
- Holistic, because is the same for all
- Passive, because we can not change it
- Ineffable, because basic formal signs (as for example the concept of set) can not be explained by other mathematical concepts

Mathematics and religion coincide in the search for certain and perennial truth. The strength of mathematics is in the argumentation. Mathematics has developed the deductive abilities of language. Mathematics helps to find common places (topoi) from which to argue, dialogue, encounter. Mathematics itself is the common place par excellence.

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