4 Text amended
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  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it 'radically erases detail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere descriptionIn a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it 'radically erases detail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it 'radically erases detail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

3 References added.
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  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it radically'radically erases detaildetail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it radically erases detail. There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it 'radically erases detail' (Sarukkai). There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

Conclusion

There is no uncontestable answer to your very good question. To the references others have given you might add :

Sundar Sarukkai, 'Applying Mathematics: The Paradoxical Relation between Mathematics, Language and Reality', Economic and Political Weekly, Vol. 38, No. 35 (Aug. 30 - Sep. 5, 2003), pp. 3662-3670.

Symposium: 'Why Are the Calculuses of Logic and Arithmetic Applicable to Reality?' G. Ryle, C. Lewy and K. R. Popper Proceedings of the Aristotelian Society, Supplementary Volumes Vol. 20, Logic and Reality (1946), pp. 20-60.

2 Text added for clarification.
source | link
  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it radically erases detail. There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it radically erases detail. There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

  1. Large portions of mathematics are of no use whatever in describing the universe. Thousands of theorems and proofs in pure mathematics are completely irrelevant to science. There is a famous Cambridge toast (whether apocryphal or not) : 'Here's to pure mathematics, and may it never be of use to anyone !'

  2. It is not mathematics that describes the world. It is physical theories which do so, and these have indispensable mathematical components but include observation, experimentation, theorising and other conceptual components which are unconnected with mathematics. In even the most successful scientific theories, mathematics is only one element.

  3. None the less it is an element, and a necessary one. Why is mathematics so important in successful science ? Its capacity for abstraction plays a part. Take the example of someone placing a ladder at an angle against a wall. Describing the situation in English or some other natural language will involve a narrative embracing the colour of the wood, the question why someone left it there, its number of rungs and whatever else comes to the narrator's mind. A mathematical description is more austere; it radically erases detail. There is a triangle formed by the ladder, the wall, and the intervening ground. From the length of the wall to the point where the ladder touches it, and the length of ground between wall and ladder we can calculate the length of the ladder, the third side of the triangle. Of course in this real-world situation the measurements will not exactly match those of the Law of Cosines applied to an ideal or model triangle. But they will be greatly more accurate than any alternative and will generalise to all triangles. In a way mathematics is so successful because it 'mathematicises' the real world; and the real world lends itself, within limits, to 'mathematicisation', to austere description. But within this situation of measurement, any number of non-mathematical, empirical assumptions are at work in the background. We assume, for instance, that the ruler by which we measure wall and ground is and remains straight, that it has been correctly calibrated, &c.

  4. This brings me to a final point, implicit in what has just been said. If mathematics plays an indispensable part in the formulation of natural laws, in few if any situations do its equations, embodied in natural laws, enable precise description or prediction outside idealised conditions. This is because the 'initial conditions' to which natural laws are applied are seldom free from 'disturbing' elements. If we apply a law describing the rate of expansion of gas a vessel containing nothing but that gas, it is unlikely that any such vessel exists. The gas will have to be absolutely pure and uncontaminated by other ingredients, and such is not the real world of scientific practice.

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