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I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1972) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) first pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent study in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1971) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent study in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1971) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) first pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent analysis in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1972) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) first pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent study in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1972) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) first pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent analysis in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1971) quite illuminating.

According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem is false – one must impose the condition of uniform convergence to get a correct statement. Lakatos (1966) first pointed out that the theorem is a perfectly correct statement about a Leibnizian continuum – an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof is correct.

The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of 1821 before it was formulated explicitly in $\epsilon-\delta$ terms by Seidel (1847) or Weierstrass.

See also a subsequent analysis in the same direction by Cutland et al. On Cauchy's notion of infinitesimal (1988).